It is currently 19 Jan 2018, 21:22

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Math: Number Theory (broken into smaller topics)

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139526 [6], given: 12794

Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

10 Mar 2010, 05:20
6
KUDOS
Expert's post
6
This post was
BOOKMARKED
NUMBER THEORY

This post is a part of [GMAT MATH BOOK]

created by: Bunuel
edited by: bb, walker

--------------------------------------------------------

The information will be included in future versions of GMAT ToolKit

--------------------------------------------------------

DEFINITION

Number Theory is concerned with the properties of numbers in general, and in particular integers.
As this is a huge issue we decided to divide it into smaller topics. Below is the list of Number Theory topics.

GMAT Number Types

GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

INTEGERS

Integers are defined as: all negative natural numbers $$\{...,-4, -3, -2, -1\}$$, zero $$\{0\}$$, and positive natural numbers $$\{1, 2, 3, 4, ...\}$$.

Note that integers do not include decimals or fractions - just whole numbers.

Even and Odd Numbers
Prime Numbers
Factors
Finding the Number of Factors of an Integer
Finding the Sum of the Factors of an Integer
Greatest Common Factor (Divisior) - GCF (GCD)
Lowest Common Multiple - LCM
Divisibility Rules
Factorials
Consecutive Integers
Evenly Spaced Set

IRRATIONAL NUMBERS

Fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333....). On the other hand, all those numbers that can be written as non-terminating, non-repeating decimals are non-rational, so they are called the "irrationals". Examples would be $$\sqrt{2}$$ ("the square root of two") or the number pi ($$\pi=$$~3.14159..., from geometry). The rationals and the irrationals are two totally separate number types: there is no overlap.

Putting these two major classifications, the rationals and the irrationals, together in one set gives you the "real" numbers.

FRACTIONS

Fractional numbers are ratios (divisions) of integers. In other words, a fraction is formed by dividing one integer by another integer. Set of Fraction is a subset of the set of Rational Numbers.

Fraction can be expressed in two forms fractional representation $$(\frac{m}{n})$$ and decimal representation $$(a.bcd)$$.

Definition
Fractional representation
Converting Improper Fractions
Reciprocal
Operation on Fractions
Decimal Representation
Converting Decimals to Fractions
Rounding
Ratios and Proportions

POSITIVE AND NEGATIVE NUMBERS

A positive number is a real number that is greater than zero.
A negative number is a real number that is smaller than zero.

Zero is not positive, nor negative.

Multiplication:
positive * positive = positive
positive * negative = negative
negative * negative = positive

Division:
positive / positive = positive
positive / negative = negative
negative / negative = positive

EXPONENTS, ROOTS, PERCENTS

Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. Roots (or radicals) are the "opposite" operation of applying exponents. A percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred").

Exponents
Perfect Square
Roots
Last Digit of a Product
Last Digit of a Power
Percents

ORDER OF OPERATIONS - PEMDAS

Perform the operations inside a Parenthesis first (absolute value signs also fall into this category), then Exponents, then Multiplication and Division, from left to right, then Addition and Subtraction, from left to right - PEMDAS.

Special cases:
• An exclamation mark indicates that one should compute the factorial of the term immediately to its left, before computing any of the lower-precedence operations, unless grouping symbols dictate otherwise. But $$3^2!$$ means $$(3^2)! = 9!$$ while $$2^{5!} = 2^{120}$$; a factorial in an exponent applies to the exponent, while a factorial not in the exponent applies to the entire power.

• If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
$$a^m^n=a^{(m^n)}$$ and not $$(a^m)^n$$.
_________________

Kudos [?]: 139526 [6], given: 12794

Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139526 [1], given: 12794

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

10 Mar 2010, 05:21
1
KUDOS
Expert's post
INTEGERS

This post is a part of [GMAT MATH BOOK]

created by: Bunuel
edited by: bb, walker

--------------------------------------------------------

The information will be included in future versions of GMAT ToolKit

--------------------------------------------------------

Definition

Integers are defined as: all negative natural numbers $$\{...,-4, -3, -2, -1\}$$, zero $$\{0\}$$, and positive natural numbers $$\{1, 2, 3, 4, ...\}$$.

Note that integers do not include decimals or fractions - just whole numbers.

Even and Odd Numbers

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder.
An even number is an integer of the form $$n=2k$$, where $$k$$ is an integer.

An odd number is an integer that is not evenly divisible by 2.
An odd number is an integer of the form $$n=2k+1$$, where $$k$$ is an integer.

Zero is an even number.

even +/- even = even;
even +/- odd = odd;
odd +/- odd = even.

Multiplication:
even * even = even;
even * odd = even;
odd * odd = odd.

Division of two integers can result into an even/odd integer or a fraction.

Prime Numbers

A Prime number is a natural number with exactly two distinct natural number divisors: 1 and itself. Otherwise a number is called a composite number. Therefore, 1 is not a prime, since it only has one divisor, namely 1. A number $$n > 1$$ is prime if it cannot be written as a product of two factors $$a$$ and $$b$$, both of which are greater than 1: n = ab.

• The first twenty-six prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

• Note: only positive numbers can be primes.

• There are infinitely many prime numbers.

• The only even prime number is 2, since any larger even number is divisible by 2. Also 2 is the smallest prime.

All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form $$6n-1$$ or $$6n+1$$, because all other numbers are divisible by 2 or 3.

• Any nonzero natural number $$n$$ can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.

Prime factorization: every positive integer greater than 1 can be written as a product of one or more prime integers in a way which is unique. For instance integer $$n$$ with three unique prime factors $$a$$, $$b$$, and $$c$$ can be expressed as $$n=a^p*b^q*c^r$$, where $$p$$, $$q$$, and $$r$$ are powers of $$a$$, $$b$$, and $$c$$, respectively and are $$\geq1$$.
Example: $$4200=2^3*3*5^2*7$$.

Verifying the primality (checking whether the number is a prime) of a given number $$n$$ can be done by trial division, that is to say dividing $$n$$ by all integer numbers smaller than $$\sqrt{n}$$, thereby checking whether $$n$$ is a multiple of $$m<\sqrt{n}$$.
Example: Verifying the primality of $$161$$: $$\sqrt{161}$$ is little less than $$13$$, from integers from $$2$$ to $$13$$, $$161$$ is divisible by $$7$$, hence $$161$$ is not prime.

• If $$n$$ is a positive integer greater than 1, then there is always a prime number $$p$$ with$$n < p < 2n$$.

Factors

A divisor of an integer $$n$$, also called a factor of $$n$$, is an integer which evenly divides $$n$$ without leaving a remainder. In general, it is said $$m$$ is a factor of $$n$$, for non-zero integers $$m$$ and $$n$$, if there exists an integer $$k$$ such that $$n = km$$.

• 1 (and -1) are divisors of every integer.

• Every integer is a divisor of itself.

• Every integer is a divisor of 0, except, by convention, 0 itself.

• Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

• A positive divisor of n which is different from n is called a proper divisor.

• An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.

• Any positive divisor of n is a product of prime divisors of n raised to some power.

• If a number equals the sum of its proper divisors, it is said to be a perfect number.
Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.

There are some elementary rules:
• If $$a$$ is a factor of $$b$$ and $$a$$ is a factor of $$c$$, then $$a$$ is a factor of $$(b + c)$$. In fact, $$a$$ is a factor of $$(mb + nc)$$ for all integers $$m$$ and $$n$$.

• If $$a$$ is a factor of $$b$$ and $$b$$ is a factor of $$c$$, then $$a$$ is a factor of $$c$$.

• If $$a$$ is a factor of $$b$$ and $$b$$ is a factor of $$a$$, then $$a = b$$ or $$a=-b$$.

• If $$a$$ is a factor of $$bc$$, and $$gcd(a,b)=1$$, then a is a factor of $$c$$.

• If $$p$$ is a prime number and $$p$$ is a factor of $$ab$$ then $$p$$ is a factor of $$a$$ or $$p$$ is a factor of $$b$$.

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Finding the Sum of the Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The sum of factors of $$n$$ will be expressed by the formula: $$\frac{(a^{p+1}-1)*(b^{q+1}-1)*(c^{r+1}-1)}{(a-1)*(b-1)*(c-1)}$$

Example: Finding the sum of all factors of 450: $$450=2^1*3^2*5^2$$

The sum of all factors of 450 is $$\frac{(2^{1+1}-1)*(3^{2+1}-1)*(5^{2+1}-1)}{(2-1)*(3-1)*(5-1)}=\frac{3*26*124}{1*2*4}=1209$$

Greatest Common Factor (Divisior) - GCF (GCD)

The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

To find the GCF, you will need to do prime-factorization. Then, multiply the common factors (pick the lowest power of the common factors).

• Every common divisor of a and b is a divisor of gcd(a, b).
• a*b=gcd(a, b)*lcm(a, b)

Lowest Common Multiple - LCM

The lowest common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero.

To find the LCM, you will need to do prime-factorization. Then multiply all the factors (pick the highest power of the common factors).

Divisibility Rules

2 - If the last digit is even, the number is divisible by 2.

3 - If the sum of the digits is divisible by 3, the number is also.

4 - If the last two digits form a number divisible by 4, the number is also.

5 - If the last digit is a 5 or a 0, the number is divisible by 5.

6 - If the number is divisible by both 3 and 2, it is also divisible by 6.

7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.

8 - If the last three digits of a number are divisible by 8, then so is the whole number.

9 - If the sum of the digits is divisible by 9, so is the number.

10 - If the number ends in 0, it is divisible by 10.

11 - If you sum every second digit and then subtract all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11.
Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11.

12 - If the number is divisible by both 3 and 4, it is also divisible by 12.

25 - Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25.

Factorials

Factorial of a positive integer $$n$$, denoted by $$n!$$, is the product of all positive integers less than or equal to n. For instance $$5!=1*2*3*4*5$$.

Trailing zeros:
Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer $$n$$, can be determined with this formula:

$$\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}$$, where k must be chosen such that $$5^k<n$$.

It's easier if you look at an example:

How many zeros are in the end (after which no other digits follow) of $$32!$$?
$$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less)

Hence, there are 7 zeros in the end of 32!

The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

Finding the number of powers of a prime number $$p$$, in the $$n!$$.

The formula is:
$$\frac{n}{p}+\frac{n}{p^2}+\frac{n}{p^3}$$ ... till $$p^x<n$$

What is the power of 2 in 25!?
$$\frac{25}{2}+\frac{25}{4}+\frac{25}{8}+\frac{25}{16}=12+6+3+1=22$$

Finding the power of non-prime in n!:

How many powers of 900 are in 50!

Make the prime factorization of the number: $$900=2^2*3^2*5^2$$, then find the powers of these prime numbers in the n!.

Find the power of 2:
$$\frac{50}{2}+\frac{50}{4}+\frac{50}{8}+\frac{50}{16}+\frac{50}{32}=25+12+6+3+1=47$$

= $$2^{47}$$

Find the power of 3:
$$\frac{50}{3}+\frac{50}{9}+\frac{50}{27}=16+5+1=22$$

=$$3^{22}$$

Find the power of 5:
$$\frac{50}{5}+\frac{50}{25}=10+2=12$$

=$$5^{12}$$

We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!.

Consecutive Integers

Consecutive integers are integers that follow one another, without skipping any integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers.

• Sum of $$n$$ consecutive integers equals the mean multiplied by the number of terms, $$n$$. Given consecutive integers $$\{-3, -2, -1, 0, 1,2\}$$, $$mean=\frac{-3+2}{2}=-\frac{1}{2}$$, (mean equals to the average of the first and last terms), so the sum equals to $$-\frac{1}{2}*6=-3$$.

• If n is odd, the sum of consecutive integers is always divisible by n. Given $$\{9,10,11\}$$, we have $$n=3$$ consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If n is even, the sum of consecutive integers is never divisible by n. Given $$\{9,10,11,12\}$$, we have $$n=4$$ consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of $$n$$ consecutive integers is always divisible by $$n!$$.
Given $$n=4$$ consecutive integers: $$\{3,4,5,6\}$$. The product of 3*4*5*6 is 360, which is divisible by 4!=24.

Evenly Spaced Set

Evenly spaced set or an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. The set of integers $$\{9,13,17,21\}$$ is an example of evenly spaced set. Set of consecutive integers is also an example of evenly spaced set.

• If the first term is $$a_1$$ and the common difference of successive members is $$d$$, then the $$n_{th}$$ term of the sequence is given by:
$$a_ n=a_1+d(n-1)$$

• In any evenly spaced set the arithmetic mean (average) is equal to the median and can be calculated by the formula $$mean=median=\frac{a_1+a_n}{2}$$, where $$a_1$$ is the first term and $$a_n$$ is the last term. Given the set $$\{7,11,15,19\}$$, $$mean=median=\frac{7+19}{2}=13$$.

• The sum of the elements in any evenly spaced set is given by:
$$Sum=\frac{a_1+a_n}{2}*n$$, the mean multiplied by the number of terms. OR, $$Sum=\frac{2a_1+d(n-1)}{2}*n$$

• Special cases:
Sum of n first integers: $$1+2+...+n=\frac{1+n}{2}*n$$

Sum of n first odd numbers: $$a_1+a_2+...+a_n=1+3+...+a_n=n^2$$, where $$a_n$$ is the last, $$n_{th}$$ term and given by: $$a_n=2n-1$$. Given $$n=5$$ first odd integers, then their sum equals to $$1+3+5+7+9=5^2=25$$.

Sum of n first positive even numbers: $$a_1+a_2+...+a_n=2+4+...+a_n=n(n+1)$$, where $$a_n$$ is the last, $$n_{th}$$ term and given by: $$a_n=2n$$. Given $$n=4$$ first positive even integers, then their sum equals to $$2+4+6+8=4(4+1)=20$$.

• If the evenly spaced set contains odd number of elements, the mean is the middle term, so the sum is middle term multiplied by number of terms. There are five terms in the set {1, 7, 13, 19, 25}, middle term is 13, so the sum is 13*5 =65.
_________________

Kudos [?]: 139526 [1], given: 12794

Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139526 [0], given: 12794

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

10 Mar 2010, 05:22
FRACTIONS

This post is a part of [GMAT MATH BOOK]

created by: Bunuel
edited by: bb, walker

--------------------------------------------------------

The information will be included in future versions of GMAT ToolKit

--------------------------------------------------------

Definition

Fractional numbers are ratios (divisions) of integers. In other words, a fraction is formed by dividing one integer by another integer. Set of Fraction is a subset of the set of Rational Numbers.

Fraction can be expressed in two forms fractional representation $$(\frac{m}{n})$$ and decimal representation $$(a.bcd)$$.

Fractional representation

Fractional representation is a way to express numbers that fall in between integers (note that integers can also be expressed in fractional form). A fraction expresses a part-to-whole relationship in terms of a numerator (the part) and a denominator (the whole).

• The number on top of the fraction is called numerator or nominator. The number on bottom of the fraction is called denominator. In the fraction, $$\frac{9}{7}$$, 9 is the numerator and 7 is denominator.

• Fractions that have a value between 0 and 1 are called proper fraction. The numerator is always smaller than the denominator. $$\frac{1}{3}$$ is a proper fraction.

• Fractions that are greater than 1 are called improper fraction. Improper fraction can also be written as a mixed number. $$\frac{5}{2}$$ is improper fraction.

• An integer combined with a proper fraction is called mixed number. $$4\frac{3}{5}$$ is a mixed number. This can also be written as an improper fraction: $$\frac{23}{5}$$

Converting Improper Fractions

• Converting Improper Fractions to Mixed Fractions:
1. Divide the numerator by the denominator
2. Write down the whole number answer
3. Then write down any remainder above the denominator
Example #1: Convert $$\frac{11}{4}$$ to a mixed fraction.
Solution: Divide $$\frac{11}{4} = 2$$ with a remainder of $$3$$. Write down the $$2$$ and then write down the remainder $$3$$ above the denominator $$4$$, like this: $$2\frac{3}{4}$$

• Converting Mixed Fractions to Improper Fractions:
1. Multiply the whole number part by the fraction's denominator
2. Add that to the numerator
3. Then write the result on top of the denominator
Example #2: Convert $$3\frac{2}{5}$$ to an improper fraction.
Solution: Multiply the whole number by the denominator: $$3*5=15$$. Add the numerator to that: $$15 + 2 = 17$$. Then write that down above the denominator, like this: $$\frac{17}{5}$$

Reciprocal

Reciprocal for a number $$x$$, denoted by $$\frac{1}{x}$$ or $$x^{-1}$$, is a number which when multiplied by $$x$$ yields $$1$$. The reciprocal of a fraction $$\frac{a}{b}$$ is $$\frac{b}{a}$$. To get the reciprocal of a number, divide 1 by the number. For example reciprocal of $$3$$ is $$\frac{1}{3}$$, reciprocal of $$\frac{5}{6}$$ is $$\frac{6}{5}$$.

Operation on Fractions

To add/subtract fractions with the same denominator, add the numerators and place that sum over the common denominator.

To add/subtract fractions with the different denominator, find the Least Common Denominator (LCD) of the fractions, rename the fractions to have the LCD and add/subtract the numerators of the fractions

Multiplying fractions: To multiply fractions just place the product of the numerators over the product of the denominators.

Dividing fractions: Change the divisor into its reciprocal and then multiply.

Example #1: $$\frac{3}{7}+\frac{2}{3}=\frac{9}{21}+\frac{14}{21}=\frac{23}{21}$$

Example #2: Given $$\frac{\frac{3}{5}}{2}$$, take the reciprocal of $$2$$. The reciprocal is $$\frac{1}{2}$$. Now multiply: $$\frac{3}{5}*\frac{1}{2}=\frac{3}{10}$$.

Decimal Representation

The decimals has ten as its base. Decimals can be terminating (ending) (such as 0.78, 0.2) or repeating (recuring) decimals (such as 0.333333....).

Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Converting Decimals to Fractions

• To convert a terminating decimal to fraction:
1. Calculate the total numbers after decimal point
2. Remove the decimal point from the number
3. Put 1 under the denominator and annex it with "0" as many as the total in step 1
4. Reduce the fraction to its lowest terms

Example: Convert $$0.56$$ to a fraction.
1: Total number after decimal point is 2.
2 and 3: $$\frac{56}{100}$$.
4: Reducing it to lowest terms: $$\frac{56}{100}=\frac{14}{25}$$

• To convert a recurring decimal to fraction:
1. Separate the recurring number from the decimal fraction
2. Annex denominator with "9" as many times as the length of the recurring number
3. Reduce the fraction to its lowest terms

Example #1: Convert $$0.393939...$$ to a fraction.
1: The recurring number is $$39$$.
2: $$\frac{39}{99}$$, the number $$39$$ is of length $$2$$ so we have added two nines.
3: Reducing it to lowest terms: $$\frac{39}{99}=\frac{13}{33}$$.

• To convert a mixed-recurring decimal to fraction:
1. Write down the number consisting with non-repeating digits and repeating digits.
2. Subtract non-repeating number from above.
3. Divide 1-2 by the number with 9's and 0's: for every repeating digit write down a 9, and for every non-repeating digit write down a zero after 9's.

Example #2: Convert $$0.2512(12)$$ to a fraction.
1. The number consisting with non-repeating digits and repeating digits is 2512;
2. Subtract 25 (non-repeating number) from above: 2512-25=2487;
3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300.

Rounding

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Ratios and Proportions

Given that $$\frac{a}{b}=\frac{c}{d}$$, where a, b, c and d are non-zero real numbers, we can deduce other proportions by simple Algebra. These results are often referred to by the names mentioned along each of the properties obtained.

$$\frac{b}{a}=\frac{d}{c}$$ - invertendo

$$\frac{a}{c}=\frac{b}{d}$$ - alternendo

$$\frac{a+b}{b}=\frac{c+d}{d}$$ - componendo

$$\frac{a-b}{b}=\frac{c-d}{d}$$ - dividendo

$$\frac{a+b}{a-b}=\frac{c+d}{c-d}$$ - componendo & dividendo
_________________

Kudos [?]: 139526 [0], given: 12794

Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139526 [2], given: 12794

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

10 Mar 2010, 05:22
2
KUDOS
Expert's post
EXPONENTS & ROOTS

This post is a part of [GMAT MATH BOOK]

created by: Bunuel
edited by: bb, walker

--------------------------------------------------------

The information will be included in future versions of GMAT ToolKit

--------------------------------------------------------

EXPONENTS

Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. For instance, number $$a$$ multiplied $$n$$ times can be written as $$a^n$$, where $$a$$ represents the base, the number that is multiplied by itself $$n$$ times and $$n$$ represents the exponent. The exponent indicates how many times to multiple the base, $$a$$, by itself.

Exponents one and zero:
$$a^0=1$$ Any nonzero number to the power of 0 is 1.
For example: $$5^0=1$$ and $$(-3)^0=1$$

$$a^1=a$$ Any number to the power 1 is itself.

Powers of zero:
If the exponent is positive, the power of zero is zero: $$0^n = 0$$, where $$n > 0$$.

If the exponent is negative, the power of zero ($$0^n$$, where $$n < 0$$) is undefined, because division by zero is implied.

Powers of one:
$$1^n=1$$ The integer powers of one are one.

Negative powers:
$$a^{-n}=\frac{1}{a^n}$$

Powers of minus one:
If n is an even integer, then $$(-1)^n=1$$.

If n is an odd integer, then $$(-1)^n =-1$$.

Operations involving the same exponents:
Keep the exponent, multiply or divide the bases
$$a^n*b^n=(ab)^n$$

$$\frac{a^n}{b^n}=(\frac{a}{b})^n$$

$$(a^m)^n=a^{mn}$$

$$a^m^n=a^{(m^n)}$$ and not $$(a^m)^n$$

Operations involving the same bases:
Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
$$a^n*a^m=a^{n+m}$$

$$\frac{a^n}{a^m}=a^{n-m}$$

Fraction as power:
$$a^{\frac{1}{n}}=\sqrt[n]{a}$$

$$a^{\frac{m}{n}}=\sqrt[n]{a^m}$$

Exponential Equations:
When solving equations with even exponents, we must consider both positive and negative possibilities for the solutions.

For instance $$a^2=25$$, the two possible solutions are $$5$$ and $$-5$$.

When solving equations with odd exponents, we'll have only one solution.

For instance for $$a^3=8$$, solution is $$a=2$$ and for $$a^3=-8$$, solution is $$a=-2$$.

Exponents and divisibility:
$$a^n-b^n$$ is ALWAYS divisible by $$a-b$$.
$$a^n-b^n$$ is divisible by $$a+b$$ if $$n$$ is even.

$$a^n + b^n$$ is divisible by $$a+b$$ if $$n$$ is odd, and not divisible by a+b if n is even.

Perfect Square

A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is an perfect square.

There are some tips about the perfect square:
• The number of distinct factors of a perfect square is ALWAYS ODD.
• The sum of distinct factors of a perfect square is ALWAYS ODD.
• A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
• Perfect square always has even number of powers of prime factors.

LAST DIGIT OF A PRODUCT

Last $$n$$ digits of a product of integers are last $$n$$ digits of the product of last $$n$$ digits of these integers.

For instance last 2 digits of 845*9512*408*613 would be the last 2 digits of 45*12*8*13=540*104=40*4=160=60

Example: The last digit of 85945*89*58307=5*9*7=45*7=35=5?

LAST DIGIT OF A POWER

Determining the last digit of $$(xyz)^n$$:

1. Last digit of $$(xyz)^n$$ is the same as that of $$z^n$$;
2. Determine the cyclicity number $$c$$ of $$z$$;
3. Find the remainder $$r$$ when $$n$$ divided by the cyclisity;
4. When $$r>0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^r$$ and when $$r=0$$, then last digit of $$(xyz)^n$$ is the same as that of $$z^c$$, where $$c$$ is the cyclisity number.

• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.
• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.
• Integers ending with 4 (eg. $$(xy4)^n$$) have a cyclisity of 2. When n is odd $$(xy4)^n$$ will end with 4 and when n is even $$(xy4)^n$$ will end with 6.
• Integers ending with 9 (eg. $$(xy9)^n$$) have a cyclisity of 2. When n is odd $$(xy9)^n$$ will end with 9 and when n is even $$(xy9)^n$$ will end with 1.

Example: What is the last digit of $$127^{39}$$?
Solution: Last digit of $$127^{39}$$ is the same as that of $$7^{39}$$. Now we should determine the cyclisity of $$7$$:

1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...

So, the cyclisity of 7 is 4.

Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of $$127^{39}$$ is the same as that of the last digit of $$7^{39}$$, is the same as that of the last digit of $$7^3$$, which is $$3$$.

ROOTS

Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.

General rules:
• $$\sqrt{x}\sqrt{y}=\sqrt{xy}$$ and $$\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}$$.

• $$(\sqrt{x})^n=\sqrt{x^n}$$

• $$x^{\frac{1}{n}}=\sqrt[n]{x}$$

• $$x^{\frac{n}{m}}=\sqrt[m]{x^n}$$

• $${\sqrt{a}}+{\sqrt{b}}\neq{\sqrt{a+b}}$$

• $$\sqrt{x^2}=|x|$$, when $$x\leq{0}$$, then $$\sqrt{x^2}=-x$$ and when $$x\geq{0}$$, then $$\sqrt{x^2}=x$$

• When the GMAT provides the square root sign for an even root, such as $$\sqrt{x}$$ or $$\sqrt[4]{x}$$, then the only accepted answer is the positive root.

That is, $$\sqrt{25}=5$$, NOT +5 or -5. In contrast, the equation $$x^2=25$$ has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.

• Odd roots will have the same sign as the base of the root. For example, $$\sqrt[3]{125} =5$$ and $$\sqrt[3]{-64} =-4$$.

• For GMAT it's good to memorize following values:
$$\sqrt{2}\approx{1.41}$$
$$\sqrt{3}\approx{1.73}$$
$$\sqrt{5}\approx{2.24}$$
$$\sqrt{6}\approx{2.45}$$
$$\sqrt{7}\approx{2.65}$$
$$\sqrt{8}\approx{2.83}$$
$$\sqrt{10}\approx{3.16}$$
_________________

Kudos [?]: 139526 [2], given: 12794

Manager
Joined: 29 Oct 2009
Posts: 194

Kudos [?]: 113 [0], given: 12

Concentration: General Management, Sustainability
WE: Consulting (Computer Software)
Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

10 Mar 2010, 06:48
This is a lucky day for me . Thank you. +1Kudos.

Kudos [?]: 113 [0], given: 12

Manager
Joined: 20 Nov 2009
Posts: 162

Kudos [?]: 264 [0], given: 64

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

02 Apr 2010, 01:42
Great resource. Thanks people!
_________________

But there’s something in me that just keeps going on. I think it has something to do with tomorrow, that there is always one, and that everything can change when it comes.
http://aimingformba.blogspot.com

Kudos [?]: 264 [0], given: 64

Intern
Joined: 29 Jun 2011
Posts: 33

Kudos [?]: 11 [0], given: 0

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

20 Jul 2011, 06:52
good stuff. thank you!

Kudos [?]: 11 [0], given: 0

Manager
Joined: 23 Aug 2011
Posts: 78

Kudos [?]: 292 [0], given: 13

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

04 Sep 2012, 20:11
Bunuel wrote:
FRACTIONS

This post is a part of [GMAT MATH BOOK]

created by: Bunuel
edited by: bb, walker

--------------------------------------------------------

The information will be included in future versions of GMAT ToolKit

--------------------------------------------------------

Definition

Fractional numbers are ratios (divisions) of integers. In other words, a fraction is formed by dividing one integer by another integer. Set of Fraction is a subset of the set of Rational Numbers.

Fraction can be expressed in two forms fractional representation $$(\frac{m}{n})$$ and decimal representation $$(a.bcd)$$.

Fractional representation

Fractional representation is a way to express numbers that fall in between integers (note that integers can also be expressed in fractional form). A fraction expresses a part-to-whole relationship in terms of a numerator (the part) and a denominator (the whole).

• The number on top of the fraction is called numerator or nominator. The number on bottom of the fraction is called denominator. In the fraction, $$\frac{9}{7}$$, 9 is the numerator and 7 is denominator.

• Fractions that have a value between 0 and 1 are called proper fraction. The numerator is always smaller than the denominator. $$\frac{1}{3}$$ is a proper fraction.

• Fractions that are greater than 1 are called improper fraction. Improper fraction can also be written as a mixed number. $$\frac{5}{2}$$ is improper fraction.

• An integer combined with a proper fraction is called mixed number. $$4\frac{3}{5}$$ is a mixed number. This can also be written as an improper fraction: $$\frac{23}{5}$$

Converting Improper Fractions

• Converting Improper Fractions to Mixed Fractions:
1. Divide the numerator by the denominator
2. Write down the whole number answer
3. Then write down any remainder above the denominator
Example #1: Convert $$\frac{11}{4}$$ to a mixed fraction.
Solution: Divide $$\frac{11}{4} = 2$$ with a remainder of $$3$$. Write down the $$2$$ and then write down the remainder $$3$$ above the denominator $$4$$, like this: $$2\frac{3}{4}$$

• Converting Mixed Fractions to Improper Fractions:
1. Multiply the whole number part by the fraction's denominator
2. Add that to the numerator
3. Then write the result on top of the denominator
Example #2: Convert $$3\frac{2}{5}$$ to an improper fraction.
Solution: Multiply the whole number by the denominator: $$3*5=15$$. Add the numerator to that: $$15 + 2 = 17$$. Then write that down above the denominator, like this: $$\frac{17}{5}$$

Reciprocal

Reciprocal for a number $$x$$, denoted by $$\frac{1}{x}$$ or $$x^{-1}$$, is a number which when multiplied by $$x$$ yields $$1$$. The reciprocal of a fraction $$\frac{a}{b}$$ is $$\frac{b}{a}$$. To get the reciprocal of a number, divide 1 by the number. For example reciprocal of $$3$$ is $$\frac{1}{3}$$, reciprocal of $$\frac{5}{6}$$ is $$\frac{6}{5}$$.

Operation on Fractions

To add/subtract fractions with the same denominator, add the numerators and place that sum over the common denominator.

To add/subtract fractions with the different denominator, find the Least Common Denominator (LCD) of the fractions, rename the fractions to have the LCD and add/subtract the numerators of the fractions

Multiplying fractions: To multiply fractions just place the product of the numerators over the product of the denominators.

Dividing fractions: Change the divisor into its reciprocal and then multiply.

Example #1: $$\frac{3}{7}+\frac{2}{3}=\frac{9}{21}+\frac{14}{21}=\frac{23}{21}$$

Example #2: Given $$\frac{\frac{3}{5}}{2}$$, take the reciprocal of $$2$$. The reciprocal is $$\frac{1}{2}$$. Now multiply: $$\frac{3}{5}*\frac{1}{2}=\frac{3}{10}$$.

Decimal Representation

The decimals has ten as its base. Decimals can be terminating (ending) (such as 0.78, 0.2) or repeating (recuring) decimals (such as 0.333333....).

Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Converting Decimals to Fractions

• To convert a terminating decimal to fraction:
1. Calculate the total numbers after decimal point
2. Remove the decimal point from the number
3. Put 1 under the denominator and annex it with "0" as many as the total in step 1
4. Reduce the fraction to its lowest terms

Example: Convert $$0.56$$ to a fraction.
1: Total number after decimal point is 2.
2 and 3: $$\frac{56}{100}$$.
4: Reducing it to lowest terms: $$\frac{56}{100}=\frac{14}{25}$$

• To convert a recurring decimal to fraction:
1. Separate the recurring number from the decimal fraction
2. Annex denominator with "9" as many times as the length of the recurring number
3. Reduce the fraction to its lowest terms

Example #1: Convert $$0.393939...$$ to a fraction.
1: The recurring number is $$39$$.
2: $$\frac{39}{99}$$, the number $$39$$ is of length $$2$$ so we have added two nines.
3: Reducing it to lowest terms: $$\frac{39}{99}=\frac{13}{33}$$.

• To convert a mixed-recurring decimal to fraction:
1. Write down the number consisting with non-repeating digits and repeating digits.
2. Subtract non-repeating number from above.
3. Divide 1-2 by the number with 9's and 0's: for every repeating digit write down a 9, and for every non-repeating digit write down a zero after 9's.

Example #2: Convert $$0.2512(12)$$ to a fraction.
1. The number consisting with non-repeating digits and repeating digits is 2512;
2. Subtract 25 (non-repeating number) from above: 2512-25=2487;
3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300.

Rounding

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Ratios and Proportions

Given that $$\frac{a}{b}=\frac{c}{d}$$, where a, b, c and d are non-zero real numbers, we can deduce other proportions by simple Algebra. These results are often referred to by the names mentioned along each of the properties obtained.

$$\frac{b}{a}=\frac{d}{c}$$ - invertendo

$$\frac{a}{c}=\frac{b}{d}$$ - alternendo

$$\frac{a+b}{b}=\frac{c+d}{d}$$ - componendo

$$\frac{a-b}{b}=\frac{c-d}{d}$$ - dividendo

$$\frac{a+b}{a-b}=\frac{c+d}{c-d}$$ - componendo & dividendo

Thanks, this a great and concise summary of all the basics.
Just to add, another important basic rule

for +ve fractions,
if $$\frac{a}{b}<1$$ and x is any positive number then $$\frac{(a+x)}{(b+x)}>\frac{a}{b}$$
,in short when the two fractions have same difference between numerator and denominator , the fraction having greater numerator is always greater.

Also, when $$\frac{a}{b}>1$$ and x is any positive number then $$\frac{(a+x)}{(b+x)}<\frac{a}{b}$$

Example which one is greater of the following two:
a)$$\frac{111487}{111490}$$ b)$$\frac{111587}{111590}$$ , the ans is b for the same reasons above, as $$\frac{111487+100}{111490+100}$$
_________________

Whatever one does in life is a repetition of what one has done several times in one's life!
If my post was worth it, then i deserve kudos

Kudos [?]: 292 [0], given: 13

Non-Human User
Joined: 09 Sep 2013
Posts: 14229

Kudos [?]: 291 [0], given: 0

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

05 Feb 2015, 23:29
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 291 [0], given: 0

Manager
Joined: 28 Jul 2011
Posts: 233

Kudos [?]: 168 [0], given: 16

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

02 Oct 2015, 09:17
• All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form 6n−1 or 6n+1, because all other numbers are divisible by 2 or 3.

Why does this property "all prime numbers above 3 are of the form 6n−1 or 6n+1" hold true for 961, but 961 is not a prime? This holds true for square of any prime number? Is there any gap in my understanding?
6n+1 = 961 --> 6n = 960

Kudos [?]: 168 [0], given: 16

Intern
Joined: 24 Jul 2015
Posts: 2

Kudos [?]: [0], given: 33

WE: Supply Chain Management (Manufacturing)
Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

03 Oct 2015, 02:57
Very useful...keep posting..

Kudos [?]: [0], given: 33

Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139526 [0], given: 12794

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

03 Oct 2015, 03:43
charan2892 wrote:
Very useful...keep posting..

Thank you.

For more check ALL YOU NEED FOR QUANT ! ! !.

Hope it helps.
_________________

Kudos [?]: 139526 [0], given: 12794

Intern
Joined: 05 Dec 2017
Posts: 1

Kudos [?]: 0 [0], given: 2

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

06 Dec 2017, 10:05
Bunuel
Thanks so much! Super useful!!

Question: The problem sets referenced at the end of number theory pdf are based on edition 12 of the GMAT official guide. I would like to try prep questions for number theory but don't have this edition of the book (I have the 2017 edition). Where can I find the questions you wrote in the pdf or where can I find questions to test the number theory concepts covered in your pdf?

Many thanks!

Kudos [?]: 0 [0], given: 2

Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139526 [1], given: 12794

Re: Math: Number Theory (broken into smaller topics) [#permalink]

### Show Tags

06 Dec 2017, 10:12
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
gmatm8 wrote:
Bunuel
Thanks so much! Super useful!!

Question: The problem sets referenced at the end of number theory pdf are based on edition 12 of the GMAT official guide. I would like to try prep questions for number theory but don't have this edition of the book (I have the 2017 edition). Where can I find the questions you wrote in the pdf or where can I find questions to test the number theory concepts covered in your pdf?

Many thanks!

2. Properties of Integers

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
_________________

Kudos [?]: 139526 [1], given: 12794

Re: Math: Number Theory (broken into smaller topics)   [#permalink] 06 Dec 2017, 10:12
Display posts from previous: Sort by


# How big can a set of integers be if all pairs have small gcd?

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?

If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ (which sends $a$ to its largest prime factor) is an injection to the primes, and so the primes themselves are the most efficient. If $M=N$ then we can take $|A|=N$. What about $M$ in between? Can you beat the primes? It would be safe to assume that $A$ consists solely of $M$-smooth numbers since we can decompose $A=A_1\cup A_2$ where each $a\in A_1$ is $M$-smooth and each $a\in A_2$ has a prime factor which at least $M+1$. The large prime factors appearing as divisors in $A_2$ cannot be repeated, so $|A_2|\leq \pi(N)$.

• Even more efficiently, you may beat the primes by adding all pairwise products of primes not exceeding $M$ (including their squares). – Ilya Bogdanov Oct 21 '15 at 16:31
• Actually, for $M= 1$ you have $|A| = \pi(N) + 1$: you can have $1$ as well as the primes. @IlyaBogdanov: there's no reason to restrict to pairwise products, you can throw in products of arbitrarily many (not necessarily distinct) primes $x = p_1 \ldots p_n$, $p_1 \le \ldots \le p_n$, as long as $x/p_1 \le M$. – Robert Israel Oct 21 '15 at 18:21
• Thus for $M=5$, in addition to the primes $\le N$ you have $\{1,4,6,8,9,10,15,25\}$. – Robert Israel Oct 21 '15 at 18:28

We'll prove that the maximal cardinality of such a set for $M^2\leq N$ has size equal to $$\pi(N)+\sum_{1<n\leq M} \pi(p(n))$$ where $p(n)$ is the smallest prime factor of $n$. Since $$\sum_{1<n\leq M} \pi(p(n))\sim \frac{M^2}{2\log^2 M},$$ this proves that Ilya Bogdanov's example of including all pairwise products of primes not exceeding $M$ is nearly optimal in terms of asymptotics.

As you suggest in the question, this problem is equivalent to constructing the largest subset $A\subset S_M(N)$ such that $\gcd(a,b)\leq M$ for every $a,b\in A$ where $S_M(N)$ is the set of $n\leq N$ whose largest prime factor is at most $M$.

To see why, suppose that $A$ satisfies $\gcd(a,b)\leq M$ for every $a,b\in A$. Then every prime that is greater than $M$ can divide at most one element of $A$. If $p>M$ divides $a\in A$, then making $a=p$ only helps create a larger set $A$. Since these primes do not interact with the $M$-smooth numbers, the proof is complete.

Let $T(N,M)$ denote the maximum size of such a set $A\subset S_M(N)$. Then the size of the largest subset of $[1,N]$ with pairwise $\gcd$'s bounded by $M$ is $$\pi(N)-\pi(M)+T(N,M).$$

As mentioned by Fedja in the comments s, the reasoning above for the primes extends to all integers. By considering those primes $p,q\leq M$ such that $pq>M$, we see that there can only be exactly one such number divisible by $pq$. Similarly for any $p,q,r$ with $pq,qr,rp\leq M$ and $pqr>M$ there can only be one number in our set divisible by $pqr$. Thus we find that the maximal size is the sum over those integers whose largest proper divisor is less than $M$. Since the largest proper divisor of $n$ equals $n/p(n)$ where $p(n)$ is the least prime factor, we can group things based on this, and we have that $$T(N,M)=\sum_{1<p\leq M}\sum_{n\leq M:\ p(n)\geq p}1.$$ Rearranging this equals $$\sum_{1<n\leq M}\sum_{p\leq p(n)}1=\sum_{n\leq M}\pi\left(p(n)\right),$$ and for composite $n$ $p(n)\leq\sqrt{n}$, so the primes dominate this sum. Thus asymptotically we have $$T(N,M)\sim\sum_{p\leq M}\pi(p)\sim\frac{M^{2}}{2\log^{2}M}.$$

• You can continue in the same spirit: consider pairs of primes $p,q\le M$ with $pq>M$. Any such pair can enter just one number, so we can as well have all pairs there. Next consider all triples such that the product of any 2 is less than or equal to $M$ but the triple product is $>M$. Again, any triple can enter at most one number, so take them all, etc. After that, of course, just add $[1,M]$. So to describe a maximal cardinality set it easy. As to the size, it looks like we just need to play with the prime number theorem carefully to figure the asymptotics but I have to teach a class now... – fedja Oct 21 '15 at 18:08
• This set can be also described as the set of all numbers whose largest proper divisor is not greater than $M$, which allows to estimate the size of the complement way faster than with my prime count idea. Now it is time to run on the stairs :-) – fedja Oct 21 '15 at 18:13
• @Fedja: Great, that works nicely! – Eric Naslund Oct 21 '15 at 18:22
• There's still something off if $M>\sqrt{N}$ which you should account for. Fedja's description of numbers with largest proper divisor not larger than $M$ is very nice. For example, this includes all numbers with no prime factors below $N/M$ which has interesting asymptotics when $M$ is larger than $\sqrt{N}$ (eventually getting to $N$ when $M=N$). So you could either restrict attention to small ranges of $M$, or perhaps flesh out the argument a little bit more ... – Lucia Oct 22 '15 at 3:19
• @Lucia: You're right, I have edited the question for now, but I'll add in a nice form for the sum when $M>\sqrt{N}$ – Eric Naslund Oct 22 '15 at 11:09


03 413 022

### Question 1

Up to now our machines have consisted of a row of boxes extending infinitely far to the left. Why not have boxes extending infinitely far to the right as well?

Let’s go back to working with a $1 \leftarrow 10$ machine and see what such boxes could mean in that machine.
(Then it should become clear what they mean in other machines as well.)

To keep the left and right boxes visibly clear, we’ll separate them with a point. (Society calls this point, for a $1 \leftarrow 10$ machine at least, a decimal point.)

So, what does it mean to have dots in the right boxes? What are the values of dots in those boxes?

Since this is a $1 \leftarrow 10$ machine, we do know that ten dots in any one box explode to make one dot one place to the left. So ten dots in the box just to the right of the decimal point are equivalent to one dot in the $1$ s box. Each dot in that box must be worth one-tenth.

We have:

In the same way, ten dots in the next box over are worth one-tenth. And so each dot in that next box must be worth one-hundredth.

We have:

And ten one-thousands make a hundredth, and ten ten-thousands make a thousandth, and so on.

We see that the boxes to the left of the decimal point represent place values as given by the powers of ten, and the boxes to the right of the decimal point place values given by the reciprocals of the powers of ten.

We have just discovered decimals!

When people write $0.3$, for example, they mean the value of placing three dots in the first box after the decimal point.

We see that $0.3$ equals three tenths: $0.3 = \dfrac{3}{10}$.

Seven dots in the third box after the decimal point is seven thousandths: $0.007 = \dfrac{7}{1000}$.

Comment: Some people might leave off the beginning zero and just write $.007 = \dfrac{7}{1000}$. It’s just a matter of personal taste.

Some people read $0.6$ out loud as “point six” and others read it out loud as “six tenths.” Which is more helpful for understanding what the number really is?

### Question 2

There is a possible source of confusion with a decimal such as $0.31$. This is technically three tenths and one hundredth: $0.31 = \dfrac{3}{10} + \dfrac{1}{100}$.

But some people read $0.31$ out loud as “thirty-one hundredths,” which looks like this.

Are these the same thing?

Well, yes! With three explosions we see that thirty-one hundredths becomes three tenths and one hundredth.

Comment: You can also show that $\dfrac{3}{10} + \dfrac{1}{100}$ and $\dfrac{31}{100}$ are the same with the arithmetic of adding fractions. We have

$\dfrac{3}{10} + \dfrac{1}{100} = \dfrac{30}{100} + \dfrac{1}{100} = \dfrac{31}{100}$.
(Do you see that this is really the result of performing three unexplosions in a picture of $\dfrac{3}{10} + \dfrac{1}{100}$?)

A teacher asked his students to each draw a $1 \leftarrow 10$ machine picture of the fraction $\dfrac{319}{1000}$.

JinJin drew:

Subra drew:

The teacher marked both students as correct. Are each of these solutions indeed valid? Explain your thinking. (By the way, the teacher doesn’t mind if students just write numbers instead of drawing dots.)

### Question 3

Multiple choice!

The decimal $0.23$ equals:

(A) $\dfrac {23}{10}$

(B) $\dfrac {23}{100}$

(C) $\dfrac {23}{1000}$

(D) $\dfrac {23}{10000}$

### Question 4

The decimal $0.0409$ equals:

(A) $\dfrac {409}{100}$

(B) $\dfrac {409}{1000}$

(C) $\dfrac {409}{10000}$

(D) $\dfrac {409}{100000}$

## SIMPLE FRACTIONS AS DECIMALS

Some decimals give fractions that simplify further.

For example,

$0.5 = \dfrac{5}{10} = \dfrac{1}{2}$

and

$0.04 = \dfrac{4}{100} = \dfrac{1}{25}$.

Conversely, if a fraction can be rewritten to have a denominator that is a power of ten, then we can easily write it as a decimal.

For example,

$\dfrac{3}{5} = \dfrac{6}{10}$ and so $\dfrac{3}{5} = 0.6$

and

$\dfrac{13}{20} = \dfrac{13 \times 5}{20 \times 5} = \dfrac{65}{100} = 0.65$.

What fractions (in simplest terms) do the following decimals represent?

$0.05$, $0.2$, $0.8$, $0.004$

### Question 6

Write each of the following fractions as a decimal.

$\dfrac {2} {5}$, $\dfrac {1} {25}$, $\dfrac {1} {20}$, $\dfrac {1} {200}$, $\dfrac {2} {2500}$

### Question 7

MULTIPLE CHOICE!

The decimal $0.050$ equals

(A)$\dfrac {50} {100}$

(B)$\dfrac {1} {20}$

(C)$\dfrac {1} {200}$

(D) None of these?

### Question 8

The decimal $0.000208$ equals

(A) $\dfrac {52} {250}$

(B) $\dfrac {52} {2500}$

(C) $\dfrac {52} {25000}$

(D) $\dfrac {52} {250000}$

### Question 9

Write each of the following fractions as decimals.

$\dfrac {7} {20}$, $\dfrac {16} {25}$, $\dfrac {301} {500}$, $\dfrac {17} {50}$, $\dfrac {3} {4}$

### Question 10

CHALLENGE

What fraction does the decimal $2.3$ represent?

### Question 11

What fraction does $17.04$ represent?

### Question 12

What fraction does $1003.1003$ represent?

### Question 13

Let’s explore the question: Do $0.19$ and $0.190$ represent the same number or different numbers?

Here are two dots and boxes pictures for the decimal $0.19$:

Here are two dots and boxes picture for the decimal $0.190$

(A) Explain how one “unexplosion” establishes that the first picture of $0.19$ is equivalent to the second picture of $0.19$.

(B) Explain how several unexplosions establishes that the first picture of $0.190$ is equivalent to the second picture of $0.190$.

(C) Explain how explosions and unexplosions in fact establish that all four pictures are equivalent to each other.

(D) In conclusion then: Does $0.190$ represent the same number as $0.19$?

### Question 14

To a mathematician, the expressions $0.19$ and $0.190$ represent exactly the same numeric quantity.

But you may have noticed in science class that scientists will often write down what seems likes
unnecessary zeros when recording measurements. This is because scientists want to impart more information to the reader than just a numeric value.

For example, suppose a botanist measures the length of a stalk. By writing the measurement as $0.190$ meters in her paper, the scientist is saying to the reader that she measured the length of the stalk to the nearest one thousandth of a meter and that she got $1$ tenth, $9$ hundredths, and $0$ thousandths of a meter. Thus we are being told that the true length of the stalk is somewhere in the range of $0.1895$ and $0.1905$ meters.

If she wrote in her paper, instead, just $0.19$ meters, then we would have to assume she measured the length of the stalk only to the nearest hundredth of a meter and so its true length lies somewhere between $0.185$ and $0.195$ meters.

## MIXED NUMBERS AS DECIMALS.

How does $12 \dfrac {3} {4}$, for example, appear as a decimal?

Well, $12 \dfrac {3} {4} = 12 + \dfrac {3} {4}$ and we can certainly write the fractional part as a decimal. (The non-fractional part is already in the $1 \leftarrow 10$ machine format!)

We have

$12 \dfrac {3} {4} = 12 + \dfrac {75} {100}$

and so we see

$12 \dfrac {3} {4} = 12.75$.

Write each of the following numbers in decimal notation.

(A) $5 \dfrac {3} {10}$

(B) $7 \dfrac {1} {5}$

(C) $13 \dfrac {1} {2}$

(D) $106 \dfrac {3} {20}$

(E) $\dfrac {78} {25}$

(F) $\dfrac {9} {4}$

(G) $\dfrac {131} {40}$

You can either play with some of the optional stations below or go to the next island!

Register NOW and unlock all islands!


# Solving $\sin(\omega t)=- \frac 1 2$

Solve the given trigonometric equation: $$\sin(\omega t)=- \frac 1 2$$

Here is my attempt:

$$\sin(\omega t)=- \frac 1 2 = \sin \biggr (\pi +\dfrac{\pi}{6}\biggr )$$

Which yields

$$\omega t = \dfrac{7\pi }{6}$$

or

$$\sin(\omega t)=- \frac 1 2 = \sin \biggr (2\pi -\dfrac{\pi}{6}\biggr )$$

$$\omega t = \dfrac{11\pi}{6}$$

Is my assumption correct?

Regards!

• I've fixed a small error. – Mr. Maxwell Jul 7 '18 at 21:17

You also have $$\omega t = \dfrac{-\pi }{6}+2k\pi$$

Thus the solution is $$\omega t = \{\dfrac{-\pi }{6}+2k_1\pi:k_1 \in Z\}\cup \{\dfrac{7\pi }{6}+2k_2\pi:k_2\in Z\}$$

Where Z is the set of integers.

From the unit circle definition of sine, we see that

$$\omega t = 2n\pi - \dfrac{\pi}{2} \pm \dfrac{\pi}{3}$$

and we can rephrase that as we wish.

The sin$(x)$ function repeats itself every $2\pi$ times, so $\omega t=\frac{7\pi}6+2kπ$ and $\omega t=\frac{11\pi}{6}+2k\pi,\;$ where $k \in \mathbb Z$.

Yes it is right but recall to add the $2k\pi$ term with $k\in \mathbb{Z}$, indeed we have

$$\sin(\omega t)=- \frac 1 2\iff \omega t=\dfrac{7\pi }{6}+2k\pi\,\lor\,\omega t=\dfrac{11\pi }{6}+2k\pi$$

• What if I don't recall to add $2k\pi$? – Mr. Maxwell Jul 7 '18 at 21:16
• @Mr.Maxwell You can lost solutions. – user Jul 7 '18 at 21:17
• However, isn't it enough to leave it up once we get $\dfrac{7\pi}{6}$ and $\dfrac{11\pi}{6}$? – Mr. Maxwell Jul 7 '18 at 21:18
• @Mr.Maxwell Those are the solutions for $\omega t \in [0,2\pi]$ but in general solving trigonometric equations we need to consider a wider range. Take a look for example here math.stackexchange.com/q/2826222/505767 – user Jul 7 '18 at 21:20
• @Mr.Maxwell - Enough is defined by who is asking the question. If the domain is not specified, then the question is poorly asked. – steven gregory Jul 7 '18 at 22:31


plot3

3-D point or line plot

Description

example

plot3(X,Y,Z) plots coordinates in 3-D space.

• To plot a set of coordinates connected by line segments, specify X, Y, and Z as vectors of the same length.

• To plot multiple sets of coordinates on the same set of axes, specify at least one of X, Y, or Z as a matrix and the others as vectors.

example

plot3(X,Y,Z,LineSpec) creates the plot using the specified line style, marker, and color.

example

plot3(X1,Y1,Z1,...,Xn,Yn,Zn) plots multiple sets of coordinates on the same set of axes. Use this syntax as an alternative to specifying multiple sets as matrices.

example

plot3(X1,Y1,Z1,LineSpec1,...,Xn,Yn,Zn,LineSpecn) assigns specific line styles, markers, and colors to each XYZ triplet. You can specify LineSpec for some triplets and omit it for others. For example, plot3(X1,Y1,Z1,'o',X2,Y2,Z2) specifies markers for the first triplet but not the for the second triplet.

example

plot3(___,Name,Value) specifies Line properties using one or more name-value pair arguments. Specify the properties after all other input arguments. For a list of properties, see Line Properties.

example

plot3(ax,___) displays the plot in the target axes. Specify the axes as the first argument in any of the previous syntaxes.

example

p = plot3(___) returns a Line object or an array of Line objects. Use p to modify properties of the plot after creating it. For a list of properties, see Line Properties.

Examples

collapse all

Define t as a vector of values between 0 and 10$\pi$. Define st and ct as vectors of sine and cosine values. Then plot st, ct, and t.

t = 0:pi/50:10*pi;
st = sin(t);
ct = cos(t);
plot3(st,ct,t)

Create two sets of x-, y-, and z-coordinates.

t = 0:pi/500:pi;
xt1 = sin(t).*cos(10*t);
yt1 = sin(t).*sin(10*t);
zt1 = cos(t);

xt2 = sin(t).*cos(12*t);
yt2 = sin(t).*sin(12*t);
zt2 = cos(t);

Call the plot3 function, and specify consecutive XYZ triplets.

plot3(xt1,yt1,zt1,xt2,yt2,zt2)

Create matrix X containing three rows of x-coordinates. Create matrix Y containing three rows of y-coordinates.

t = 0:pi/500:pi;
X(1,:) = sin(t).*cos(10*t);
X(2,:) = sin(t).*cos(12*t);
X(3,:) = sin(t).*cos(20*t);

Y(1,:) = sin(t).*sin(10*t);
Y(2,:) = sin(t).*sin(12*t);
Y(3,:) = sin(t).*sin(20*t);

Create matrix Z containing the z-coordinates for all three sets.

Z = cos(t);

Plot all three sets of coordinates on the same set of axes.

plot3(X,Y,Z)

Create vectors xt, yt, and zt.

t = 0:pi/500:40*pi;
xt = (3 + cos(sqrt(32)*t)).*cos(t);
yt = sin(sqrt(32) * t);
zt = (3 + cos(sqrt(32)*t)).*sin(t);

Plot the data, and use the axis equal command to space the tick units equally along each axis. Then specify the labels for each axis.

plot3(xt,yt,zt)
axis equal
xlabel('x(t)')
ylabel('y(t)')
zlabel('z(t)')

Create vectors t, xt, and yt, and plot the points in those vectors using circular markers.

t = 0:pi/20:10*pi;
xt = sin(t);
yt = cos(t);
plot3(xt,yt,t,'o')

Create vectors t, xt, and yt, and plot the points in those vectors as a blue line with 10-point circular markers. Use a hexadecimal color code to specify a light blue fill color for the markers.

t = 0:pi/20:10*pi;
xt = sin(t);
yt = cos(t);
plot3(xt,yt,t,'-o','Color','b','MarkerSize',10,'MarkerFaceColor','#D9FFFF')

Create vector t. Then use t to calculate two sets of x and y values.

t = 0:pi/20:10*pi;
xt1 = sin(t);
yt1 = cos(t);

xt2 = sin(2*t);
yt2 = cos(2*t);

Plot the two sets of values. Use the default line for the first set, and specify a dashed line for the second set.

plot3(xt1,yt1,t,xt2,yt2,t,'--')

Create vectors t, xt, and yt, and plot the data in those vectors. Return the chart line in the output variable p.

t = linspace(-10,10,1000);
xt = exp(-t./10).*sin(5*t);
yt = exp(-t./10).*cos(5*t);
p = plot3(xt,yt,t);

Change the line width to 3.

p.LineWidth = 3;

Starting in R2019b, you can display a tiling of plots using the tiledlayout and nexttile functions. Call the tiledlayout function to create a 1-by-2 tiled chart layout. Call the nexttile function to create the axes objects ax1 and ax2. Create separate line plots in the axes by specifying the axes object as the first argument to plot3.

tiledlayout(1,2)

% Left plot
ax1 = nexttile;
t = 0:pi/20:10*pi;
xt1 = sin(t);
yt1 = cos(t);
plot3(ax1,xt1,yt1,t)
title(ax1,'Helix With 5 Turns')

% Right plot
ax2 = nexttile;
t = 0:pi/20:10*pi;
xt2 = sin(2*t);
yt2 = cos(2*t);
plot3(ax2,xt2,yt2,t)
title(ax2,'Helix With 10 Turns')

Create x and y as vectors of random values between 0 and 1. Create z as a vector of random duration values.

x = rand(1,10);
y = rand(1,10);
z = duration(rand(10,1),randi(60,10,1),randi(60,10,1));

Plot x, y, and z, and specify the format for the z-axis as minutes and seconds. Then add axis labels, and turn on the grid to make it easier to visualize the points within the plot box.

plot3(x,y,z,'o','DurationTickFormat','mm:ss')
xlabel('X')
ylabel('Y')
zlabel('Duration')
grid on

Create vectors xt, yt, and zt. Plot the values, specifying a solid line with circular markers using the LineSpec argument. Specify the MarkerIndices property to place one marker at the 200th data point.

t = 0:pi/500:pi;
xt(1,:) = sin(t).*cos(10*t);
yt(1,:) = sin(t).*sin(10*t);
zt = cos(t);
plot3(xt,yt,zt,'-o','MarkerIndices',200)

Input Arguments

collapse all

x-coordinates, specified as a scalar, vector, or matrix. The size and shape of X depends on the shape of your data and the type of plot you want to create. This table describes the most common situations.

Type of PlotHow to Specify Coordinates
Single point

Specify X, Y, and Z as scalars and include a marker. For example:

plot3(1,2,3,'o')

One set of points

Specify X, Y, and Z as any combination of row or column vectors of the same length. For example:

plot3([1 2 3],[4; 5; 6],[7 8 9])

Multiple sets of points
(using vectors)

Specify consecutive sets of X, Y, and Z vectors. For example:

plot3([1 2 3],[4 5 6],[7 8 9],[1 2 3],[4 5 6],[10 11 12])

Multiple sets of points
(using matrices)

Specify at least one of X, Y, or Z as a matrix, and the others as vectors. Each of X, Y, and Z must have at least one dimension that is same size. For best results, specify all vectors of the same shape and all matrices of the same shape. For example:

plot3([1 2 3],[4 5 6],[7 8 9; 10 11 12])

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical | datetime | duration

y-coordinates, specified as a scalar, vector, or matrix. The size and shape of Y depends on the shape of your data and the type of plot you want to create. This table describes the most common situations.

Type of PlotHow to Specify Coordinates
Single point

Specify X, Y, and Z as scalars and include a marker. For example:

plot3(1,2,3,'o')

One set of points

Specify X, Y, and Z as any combination of row or column vectors of the same length. For example:

plot3([1 2 3],[4; 5; 6],[7 8 9])

Multiple sets of points
(using vectors)

Specify consecutive sets of X, Y, and Z vectors. For example:

plot3([1 2 3],[4 5 6],[7 8 9],[1 2 3],[4 5 6],[10 11 12])

Multiple sets of points
(using matrices)

Specify at least one of X, Y, or Z as a matrix, and the others as vectors. Each of X, Y, and Z must have at least one dimension that is same size. For best results, specify all vectors of the same shape and all matrices of the same shape. For example:

plot3([1 2 3],[4 5 6],[7 8 9; 10 11 12])

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical | datetime | duration

z-coordinates, specified as a scalar, vector, or matrix. The size and shape of Z depends on the shape of your data and the type of plot you want to create. This table describes the most common situations.

Type of PlotHow to Specify Coordinates
Single point

Specify X, Y, and Z as scalars and include a marker. For example:

plot3(1,2,3,'o')

One set of points

Specify X, Y, and Z as any combination of row or column vectors of the same length. For example:

plot3([1 2 3],[4; 5; 6],[7 8 9])

Multiple sets of points
(using vectors)

Specify consecutive sets of X, Y, and Z vectors. For example:

plot3([1 2 3],[4 5 6],[7 8 9],[1 2 3],[4 5 6],[10 11 12])

Multiple sets of points
(using matrices)

Specify at least one of X, Y, or Z as a matrix, and the others as vectors. Each of X, Y, and Z must have at least one dimension that is same size. For best results, specify all vectors of the same shape and all matrices of the same shape. For example:

plot3([1 2 3],[4 5 6],[7 8 9; 10 11 12])

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical | datetime | duration

Line style, marker, and color, specified as a character vector or string containing symbols. The symbols can appear in any order. You do not need to specify all three characteristics (line style, marker, and color). For example, if you omit the line style and specify the marker, then the plot shows only the marker and no line.

Example: '--or' is a red dashed line with circle markers

Line StyleDescription
-Solid line (default)
--Dashed line
:Dotted line
-.Dash-dot line
MarkerDescription
oCircle
+Plus sign
*Asterisk
.Point
xCross
sSquare
dDiamond
^Upward-pointing triangle
vDownward-pointing triangle
>Right-pointing triangle
<Left-pointing triangle
pPentagram
hHexagram
ColorDescription

y

yellow

m

magenta

c

cyan

r

red

g

green

b

blue

w

white

k

black

Target axes, specified as an Axes object. If you do not specify the axes and if the current axes is Cartesian, then plot3 uses the current axes.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: plot3([1 2],[3 4],[5 6],'Color','red') specifies a red line for the plot.

Note

The properties listed here are only a subset. For a complete list, see Line Properties.

Color, specified as an RGB triplet, a hexadecimal color code, a color name, or a short name. The color you specify sets the line color. It also sets the marker edge color when the MarkerEdgeColor property is set to 'auto'.

For a custom color, specify an RGB triplet or a hexadecimal color code.

• An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of the color. The intensities must be in the range [0,1]; for example, [0.4 0.6 0.7].

• A hexadecimal color code is a character vector or a string scalar that starts with a hash symbol (#) followed by three or six hexadecimal digits, which can range from 0 to F. The values are not case sensitive. Thus, the color codes '#FF8800', '#ff8800', '#F80', and '#f80' are equivalent.

Alternatively, you can specify some common colors by name. This table lists the named color options, the equivalent RGB triplets, and hexadecimal color codes.

Color NameShort NameRGB TripletHexadecimal Color CodeAppearance
'red''r'[1 0 0]'#FF0000'

'green''g'[0 1 0]'#00FF00'

'blue''b'[0 0 1]'#0000FF'

'cyan' 'c'[0 1 1]'#00FFFF'

'magenta''m'[1 0 1]'#FF00FF'

'yellow''y'[1 1 0]'#FFFF00'

'black''k'[0 0 0]'#000000'

'white''w'[1 1 1]'#FFFFFF'

'none'Not applicableNot applicableNot applicableNo color

Here are the RGB triplets and hexadecimal color codes for the default colors MATLAB® uses in many types of plots.

[0 0.4470 0.7410]'#0072BD'

[0.8500 0.3250 0.0980]'#D95319'

[0.9290 0.6940 0.1250]'#EDB120'

[0.4940 0.1840 0.5560]'#7E2F8E'

[0.4660 0.6740 0.1880]'#77AC30'

[0.3010 0.7450 0.9330]'#4DBEEE'

[0.6350 0.0780 0.1840]'#A2142F'

Line width, specified as a positive value in points, where 1 point = 1/72 of an inch. If the line has markers, then the line width also affects the marker edges.

The line width cannot be thinner than the width of a pixel. If you set the line width to a value that is less than the width of a pixel on your system, the line displays as one pixel wide.

Marker size, specified as a positive value in points, where 1 point = 1/72 of an inch.

Marker outline color, specified as 'auto', an RGB triplet, a hexadecimal color code, a color name, or a short name. The default value of 'auto' uses the same color as the Color property.

For a custom color, specify an RGB triplet or a hexadecimal color code.

• An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of the color. The intensities must be in the range [0,1]; for example, [0.4 0.6 0.7].

• A hexadecimal color code is a character vector or a string scalar that starts with a hash symbol (#) followed by three or six hexadecimal digits, which can range from 0 to F. The values are not case sensitive. Thus, the color codes '#FF8800', '#ff8800', '#F80', and '#f80' are equivalent.

Alternatively, you can specify some common colors by name. This table lists the named color options, the equivalent RGB triplets, and hexadecimal color codes.

Color NameShort NameRGB TripletHexadecimal Color CodeAppearance
'red''r'[1 0 0]'#FF0000'

'green''g'[0 1 0]'#00FF00'

'blue''b'[0 0 1]'#0000FF'

'cyan' 'c'[0 1 1]'#00FFFF'

'magenta''m'[1 0 1]'#FF00FF'

'yellow''y'[1 1 0]'#FFFF00'

'black''k'[0 0 0]'#000000'

'white''w'[1 1 1]'#FFFFFF'

'none'Not applicableNot applicableNot applicableNo color

Here are the RGB triplets and hexadecimal color codes for the default colors MATLAB uses in many types of plots.

[0 0.4470 0.7410]'#0072BD'

[0.8500 0.3250 0.0980]'#D95319'

[0.9290 0.6940 0.1250]'#EDB120'

[0.4940 0.1840 0.5560]'#7E2F8E'

[0.4660 0.6740 0.1880]'#77AC30'

[0.3010 0.7450 0.9330]'#4DBEEE'

[0.6350 0.0780 0.1840]'#A2142F'

Marker fill color, specified as 'auto', an RGB triplet, a hexadecimal color code, a color name, or a short name. The 'auto' option uses the same color as the Color property of the parent axes. If you specify 'auto' and the axes plot box is invisible, the marker fill color is the color of the figure.

For a custom color, specify an RGB triplet or a hexadecimal color code.

• An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of the color. The intensities must be in the range [0,1]; for example, [0.4 0.6 0.7].

• A hexadecimal color code is a character vector or a string scalar that starts with a hash symbol (#) followed by three or six hexadecimal digits, which can range from 0 to F. The values are not case sensitive. Thus, the color codes '#FF8800', '#ff8800', '#F80', and '#f80' are equivalent.

Alternatively, you can specify some common colors by name. This table lists the named color options, the equivalent RGB triplets, and hexadecimal color codes.

Color NameShort NameRGB TripletHexadecimal Color CodeAppearance
'red''r'[1 0 0]'#FF0000'

'green''g'[0 1 0]'#00FF00'

'blue''b'[0 0 1]'#0000FF'

'cyan' 'c'[0 1 1]'#00FFFF'

'magenta''m'[1 0 1]'#FF00FF'

'yellow''y'[1 1 0]'#FFFF00'

'black''k'[0 0 0]'#000000'

'white''w'[1 1 1]'#FFFFFF'

'none'Not applicableNot applicableNot applicableNo color

Here are the RGB triplets and hexadecimal color codes for the default colors MATLAB uses in many types of plots.

[0 0.4470 0.7410]'#0072BD'

[0.8500 0.3250 0.0980]'#D95319'

[0.9290 0.6940 0.1250]'#EDB120'

[0.4940 0.1840 0.5560]'#7E2F8E'

[0.4660 0.6740 0.1880]'#77AC30'

[0.3010 0.7450 0.9330]'#4DBEEE'

[0.6350 0.0780 0.1840]'#A2142F'

Tips

• Use NaN or Inf to create breaks in the lines. For example, this code plots a line with a break between z=2 and z=4.

plot3([1 2 3 4 5],[1 2 3 4 5],[1 2 NaN 4 5])

• plot3 uses colors and line styles based on the ColorOrder and LineStyleOrder properties of the axes. plot3 cycles through the colors with the first line style. Then, it cycles through the colors again with each additional line style.

Starting in R2019b, you can change the colors and the line styles after plotting by setting the ColorOrder or LineStyleOrder properties on the axes. You can also call the colororder function to change the color order for all the axes in the figure.


# Math Help - Integrate LN2x/x^2

1. ## Integrate LN2x/x^2

Hey guys, thanks for all the help so far.. stuck on another problem.

$\int \frac{ln2x}{x^2}$

So I think
$ln2x$
is an integration by parts problem by itself, but I am lost.
I know that
$\int lnx = xlnx-x$
But that was just a rule given by the professor, not sure how we applied that.

I think it goes something like this (using integration by parts)
$u= ln2x \ du= \frac{2x}{x} \ dv=dx \ v=x$

Using the rule
$UV- \int VdU$

I get
$(xln2x)- \int x \frac{2x}{x}$
which equals
$(xln2x)- \int \frac{2x^2}{x}$

But that's just for the top portion of the original and I am not even sure I did that correctly.

Thanks guys

2. Just in case a picture helps...

... where (key in first spoiler) ...

Spoiler:

... is the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x. And,

... is lazy integration by parts, doing without u and v.

Spoiler:

________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

3. Originally Posted by Spoolx
Hey guys, thanks for all the help so far.. stuck on another problem.

$\int \frac{ln2x}{x^2}$

So I think
$ln2x$
is an integration by parts problem by itself, but I am lost.
I know that
$\int lnx = xlnx-x$
But that was just a rule given by the professor, not sure how we applied that.

I think it goes something like this (using integration by parts)
$u= ln2x \ du= \frac{2x}{x} \ dv=dx \ v=x$

Using the rule
$UV- \int VdU$

I get
$(xln2x)- \int x \frac{2x}{x}$
which equals
$(xln2x)- \int \frac{2x^2}{x}$

But that's just for the top portion of the original and I am not even sure I did that correctly.

Thanks guys
$ln(2x)=ln2+lnx$

$\displaystyle\int{\frac{ln(2x)}{x^2}}dx=ln2\int{\f rac{1}{x^2}}dx+\int{\frac{lnx}{x^2}}dx$

Now you can use integration by parts.

4. Hello, Spoolx!

$\displaystyle \int \frac{\ln(2x)}{x^2}\,dx$

Integrate by parts:

. . $\begin{array}{cccccccc}
u &=& \ln(2x) && dv &=& \dfrac{1}{x^2}\,dx \\ \\[-3mm]
du &=& \dfrac{dx}{x} && v &=& \text{-}\dfrac{1}{x} \end{array}$

And we have: . $\displaystyle \text{-}\frac{1}{x}\ln(2x) + \int\frac{dx}{x^2}$

. . . . . . . . $\displaystyle =\;\text{-}\frac{1}{x}\ln(2x) - \frac{1}{x} + C$

. . . . . . . . $\displaystyle =\;\text{-}\frac{1}{x}\left[\ln(2x) + 1\right] + C$

5. Thanks for the help guys, can you jst run through the derivation of the $ln2x$ please?

6. Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case ), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

The general drift is...

Though, as Archie Meade points out, you can split the log first to avoid the chain rule.

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

7. Originally Posted by Spoolx
Hey guys, thanks for all the help so far.. stuck on another problem.

$\int \frac{ln2x}{x^2}$

So I think
$ln2x$
is an integration by parts problem by itself, but I am lost.
I know that
$\int lnx = xlnx-x$
But that was just a rule given by the professor, not sure how we applied that.

I think it goes something like this (using integration by parts)
$u= ln2x \ du= \frac{2x}{x} \ dv=dx \ v=x$

Using the rule
$UV- \int VdU$

I get
$(xln2x)- \int x \frac{2x}{x}$
which equals
$(xln2x)- \int \frac{2x^2}{x}$

But that's just for the top portion of the original and I am not even sure I did that correctly.

Thanks guys
Rather than trying to apply integration by parts just to the numerator,
you should try applying the technique to the entire expression.

$\displaystyle\int{u}dv=uv-\int{v}du$

$u=ln(2x),\;\;dv=\displaystyle\frac{1}{x^2}dx$

We need $v$ and $du$ to apply integration by parts. Using the chain rule we get

$\displaystyle\frac{du}{dx}=\frac{1}{2x}2$

$\Rightarrow\ du=\frac{1}{x}dx$

$v=\int{x^{-2}}dx=-x^{-1}$

$\displaystyle\ uv-\int{v}du}=-\frac{ln(2x)}{x}+\int{\frac{1}{x^2}dx=-\frac{ln(2x)}{x}-\frac{1}{x}+C$

Alternatively, begin with

$ln(2x)=ln2+lnx$

8. Glad I found this forum, learning alot from you guys!
Thanks!


# Math Help - verifying answers for polynomials

1. ## verifying answers for polynomials

I would like to verify that the answers that I have come up with are correct on the following 3 equations. I greatly appreciate it! (the numbers in red are exponents) 1) solve the equation 1/4a2 + 3/4 a=1 My answer is a=4 and a=1 ??? 2) Simplify the expression 2ab4 - 3 a2b2 - ab4 + a2b2 I have ab4 - 2a2b2 ??? and 3) Simplify the expression -2(4y2+3z3+5) + 3(2y2-5z3+3) My answer is -21z3 - 2y2 -1??? I thank you for the help!

2. ## latex

Originally Posted by jay1
I would like to verify that the answers that I have come up with are correct on the following 3 equations. I greatly appreciate it! (the numbers in red are exponents) 1) solve the equation 1/4a2 + 3/4 a=1 My answer is a=4 and a=1 ??? 2) Simplify the expression 2ab4 - 3 a2b2 - ab4 + a2b2 I have ab4 - 2a2b2 ??? and 3) Simplify the expression -2(4y2+3z3+5) + 3(2y2-5z3+3) My answer is -21z3 - 2y2 -1??? I thank you for the help!
hope you learn to use laTex
this is my understanding of your equation

$

\frac{1}{4a^2} + \frac{3}{4a} = 1
$

$\frac{1}{4a^2} + \frac{3}{4a}\times\frac{a}{a} \Rightarrow \frac{1+3a}{4a^2} = 1$

$4a^2 - 3a -1 = 0$

$(4a-1)(a+1)= 0$

$a= -1$

$a=-\frac{1}{4}$

3. Is there a place that will show me how to use this "laTex"? BigWave, was my answer correct? Thanks for your help.

4. Originally Posted by jay1
I would like to verify that the answers that I have come up with are correct on the following 3 equations. I greatly appreciate it! (the numbers in red are exponents) 1) solve the equation 1/4a2 + 3/4 a=1 My answer is a=4 and a=1 ??? 2) Simplify the expression 2ab4 - 3 a2b2 - ab4 + a2b2 I have ab4 - 2a2b2 ??? and 3) Simplify the expression -2(4y2+3z3+5) + 3(2y2-5z3+3) My answer is -21z3 - 2y2 -1??? I thank you for the help!

If problem 1 is what I interpreted it to be, which is this:

$
\frac{1}{4a^2} + \frac{3}{4}a = 1
$

then

$
a=1, a=\frac{1+\sqrt{13}}{6}, \text{ and } a=\frac{1-\sqrt{13}}{6}
$
.

However, if problem 1 is this:

$
\frac{1}{4a^2} + \frac{3}{4a} = 1
$

then

$
a=1, a= -\frac{1}{4}
$
.

And if problem 1 is this:

$
\frac{1}{4}a^2 + \frac{3}{4}a = 1
$

then

$
a=1, a= -4
$
.

5. Hello, jay1!

1) Solve: . $\tfrac{1}{4}a^2 + \tfrac{3}{4}a \:=\:1$

My answer: . $a=4,\;a=1$ . . . . no
We have: . $\tfrac{1}{4}a^2 + \tfrac{3}{4}a \:=\:1$

Multiply by 4: . $a^2 + 3a \:=\:4 \quad\Rightarrow\quad a^2 + 3a - 4 \:=\:0$

Factor: . $(a-1)(a+4) \:=\:0$

Therefore: . $a\;=\;1,\:-4$

2) Simplify: . $2ab^4 -3a^2b^2 - ab^4 + a^2b^2$

I have: . $ab^4- 2a^2b^2$ . . . . Yes!

3) Simplify: . $-2(4y^2 +3z^3 +5) + 3(2y^2 -5z^3 +3)$

My answer: . $-21z^3 - 2y^2 -1$ . . . . Right!

6. PROBLEM 2:

$
2ab^4 - 3 a^2b^2 - ab^4 + a^2b^2 = ab^4 -2a^2b^2
$

You are correct.

7. PROBLEM 3:

$
-2(4y2+3z3+5) + 3(2y2-5z3+3) = (-8y^2 - 6z^3 - 10) + (6y^2 - 15z^3 + 9)$
$= -8y^2 - 6z^3 - 10 + 6y^2 - 15z^3 + 9 = -21z^3 - 2y^2 - 1$

Good job again. Problem 3 is correct.

-Andy

8. ## mystery equation

who had the mystery equation correct....
just curious..

9. Thanks for all of the help. I have a few more that I would like confirmation/correction for (exponents are denoted in red font). It helps to know if I am on the right track. 1) solve the equation x2 + 4x - 45 = 0 I have x=9 and x=-5 ??? 2) find the product of (x-2y)2 I have x2 - 4xy + 4x2 ??? 3) completely factor the expression y2 + 12y + 35 I have come up with (y+7) (y-5)?

10. BigWave, I think that it was Soroban. Thanks again!

11. Originally Posted by jay1
Thanks for all of the help. I have a few more that I would like confirmation/correction for (exponents are denoted in red font). It helps to know if I am on the right track. 1) solve the equation x2 + 4x - 45 = 0 I have x=9 and x=-5 ??? 2) find the product of (x-2y)2 I have x2 - 4xy + 4x2 ??? 3) completely factor the expression y2 + 12y + 35 I have come up with (y+7) (y-5)?

$x^2 + 4x - 45 = 0$

$(x+9)(x-5) = 0$

$x = 5, -9$

12. Originally Posted by jay1
2) Simplify the expression 2ab4 - 3 a2b2 - ab4 + a2b2 [COLOR=black]I have ab4 - 2a2b2

$
2ab^4 - 3a^2b^2 -ab^4 + a^2b^2 = ab^4 - 2a^2b^2
$

Atta boy (assuming you are a guy).

13. 3) completely factor the expression y2 + 12y + 35 I have come up with (y+7) (y-5)?

$
y^2 + 12y + 35 = (y+7)(y+5) \implies y=-5,-7
$

Good luck.

-Andy

14. Perform the indicated operations on this expression: (5a^3 + 3a -2) - (4a^3 + a^2 + 5) I have a^3 + a^2 + 3 Is this right?

15. Originally Posted by jay1
Perform the indicated operations on this expression: (5a^3 + 3a -2) - (4a^3 + a^2 + 5) I have a^3 + a^2 + 3 Is this right?
No.

$
(5a^3 + 3a -2) - (4a^3 + a^2 + 5) = 5a^3 + 3a - 2 - 4a^3 - a^2 - 5 =
a^3 - a^2 + 3a - 7
$

The minus before the parenthesis changes the sign of each term inside the parentheses.

-Andy

Page 1 of 2 12 Last


How can I simplify this fraction problem?

I have the problem $$\frac{x^2}{x^2-4} - \frac{x+1}{x+2}$$ which should simplify to $$\frac{1}{x-2}$$

I have simplified $$x^2-4$$, which becomes:

$$\frac{x^2}{(x-2)(x+2)} - \frac{x+1}{x+2}$$

However, if I combine the fractions I get, $$x^2-x-1$$ for the numerator, which can't be factored. That's where I get stuck.

How can I get $$\frac{1}{x-2}$$ out of this problem?

You need to put them over a common denominator, $$\frac {x+1}{x+2}=\frac {(x+1)(x-2)}{(x+2)(x-2)}=\frac {x^2-x-2}{x^2-4}$$ Now you can subtract the numerators $$x^2-(x^2-x-2)=x+2$$ and finally divide out the $$x+2$$ from numerator and denominator

Your $$-1$$ should be $$-2$$. You didn't show your work, so I can't see why it happened.

• Regarding your last line, I wonder if OP just added the two numerators together without finding a common denominator. – Matthew Leingang Jan 30 at 18:54
• @MatthewLeingang That's right. It's been awhile since I worked with fractions, and I'm very rusty. – LuminousNutria Jan 30 at 18:56
• @LuminousNutria It's a very common error. But if you think about it with familiar fractions you'll remember it can't be true. For instance, $\frac{1}{2} + \frac{1}{2}$ is $1$, not $\frac{2}{4}$ (which is $\frac{1}{2}$ again). – Matthew Leingang Jan 30 at 19:00

\begin{align} \frac{x^2}{(x-2)(x+2)} - \frac{x+1}{x+2}&=\frac1{x+2}\left(\frac{x^2}{x-2} - (x+1)\right)\\ &=\frac1{x+2}\left(\frac{x^2-(x-2)(x+1)}{x-2}\right)\\ &=\frac1{x+2}\left(\frac{x^2-(x^2-x-2)}{x-2}\right)\\ &=\frac1{x+2}\left(\frac{x+2}{x-2}\right)\\ &=\frac1{x-2} \end{align}

Write $$\frac{x^2}{(x-2)(x+2)}-\frac{x+1}{x+2}=\frac{x^2}{(x-2)(x+2)}-\frac{(x+1)(x-2)}{(x+2)(x-2)}=…$$ Note that it must be $$x\ne 2,-2$$

The numerator should be $$x^2-(x-2)(x+1) = x+2$$ which simplifies with a factor of the denominator.

You probably just forgot a term when subtracting the two fractions.


# Prove that for all integers $a$ and $b$, if $a$ divides $b$, then $a^{2}$ divides $b^{2}$.

I just need to know that if $a$ divides $b$, where $a$ and $b$ are integers, does $a^{2}$ divide $b^{2}$?

-

If $a$ divides $b$, then $b=ka$ for some integer $k$, so $b^2=k^2a^2$ where $k^2$ is an integer.

-
How do you prove that k^2 is an integer? Is that by closure? –  amster27 Feb 22 '13 at 1:07
@amster27: If you are concerned about whether $k^{2}$ is an integer, then you should be equally concerned about whether both $a^{2}$ and $b^{2}$ are integers. :) –  Haskell Curry Feb 22 '13 at 1:23
Thanks, I'm just starting to write proofs in a Mathematical Reasoning class, and I just feel so lost. But these explanations helped. –  amster27 Feb 22 '13 at 1:28
@amster: Note that $+_{\mathbb{Z}}: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ and $\times_{\mathbb{Z}}: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$. Therefore, adding two integers or multiplying two integers yields an integer. –  Haskell Curry Feb 22 '13 at 1:44

Hint $\rm\ \ a\mid b\ \Rightarrow\ \dfrac{b}a\in \Bbb Z\ \Rightarrow\ \dfrac{b^2}{a^2} = \left(\dfrac{b}a\right)^2\!\in \Bbb Z^2\subseteq \Bbb Z\:\Rightarrow\: a^2\mid b^2\$

-
That's closer to a proof than a hint. –  1015 Feb 22 '13 at 1:29
The argument is elegant, but I think that the OP wants to stay within $\mathbb{Z}$ and does not wish to jump into $\mathbb{Q}$, as he seems to be very interested in the axioms governing the properties of $\mathbb{Z}$. –  Haskell Curry Feb 22 '13 at 1:29
@Haskell Possibly, but there was no hint of any such constraint in the question. –  Math Gems Feb 22 '13 at 1:46