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# There are 1600 jelly beans divided between two jars, X and Y. If the

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There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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06 Feb 2015, 09:03
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There are 1600 jelly beans divided between two jars, X and Y. If there are 100 fewer jelly beans in jar X than three times the number of beans in jar Y, how many beans are in jar X?

A. 375

B. 950

C. 1150

D. 1175

E. 1350
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Re: There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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06 Feb 2015, 13:43
4
1
Hi All,

This question can certainly be solved with Algebra; with 2 variables and 2 unique equations, it's just a matter of translating the text into equations and doing "system" math.

The answers are numbers though, and there is a logical pattern in the prompt that you can use to quickly TEST THE ANSWERS.

We're told that there are a total of 1600 marbles in two jars. Jar X has 100 less than THREE TIMES the number of marbles in Jar Y. We're asked for the number of marbles in Jar X.

100 marbles is a relatively small amount, compared to the 1600 total marbles. If we ignore the "100" and do a rough estimation, Jar X would be about 1200 and Jar Y would be about 400 (which matches the information about THREE TIMES the number of marbles). Since we DO have to factor in the 100 marbles though, the number of marbles in X has to be LESS than 1200. With this deduction, the answer has to be C or D.

Let's TEST C (since it looks like the "nicer" number to deal with).

IF....
X = 1150 marbles
X + 100 = 1250
1250/3 = 416.6666 marbles
1150 + 416.66666 is NOT 1600 total marbles.
Eliminate C.

Here's the proof though:

IF....
X = 1175 marbles
X + 100 = 1275
1275/3 = 425
1175 + 425 = 1600 marbles

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Re: There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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06 Feb 2015, 13:28
1
x+y=1600 , X=1600-Y
x+100=3y, 1700=4Y
Y=425

x=1600-y=1600-425=1175
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Re: There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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06 Nov 2018, 20:08
pacifist85 wrote:
There are 1600 jelly beans divided between two jars, X and Y. If there are 100 fewer jelly beans in jar X than three times the number of beans in jar Y, how many beans are in jar X?

A. 375

B. 950

C. 1150

D. 1175

E. 1350

Let Jelly beans in jar $$X = x$$ and $$Y = y$$

$$x + y = 1600$$ ----- ($$i$$)

Given jar $$X$$ has $$100$$ fewer jelly beans than three times the number of beans in jar $$Y$$.

Therefore; $$x + 100 = 3y$$

$$x = 3y - 100$$

Substituting value of $$x$$ in equation ($$i$$), we get;

$$3y - 100 + y = 1600$$

$$4y = 1600 + 100 = 1700$$

$$y = \frac{1700}{4} = 425$$

Substituting value of $$y$$ in equation ($$i$$), we get;

$$x + 425 = 1600$$

$$x = 1600$$ $$-$$ $$425 = 1175$$

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Re: There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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08 Mar 2019, 17:52
Hi,
If X has 100 fewer jelly beans than three times the number of beans in jar Y.
Then why did we write X+100 ?
Shouldnt it be X-100?
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Re: There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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08 Mar 2019, 19:35
x+y=1600 (I)

And

3*y = x-100

Reorganizing it

3*y-100 = x (II)

Subs. (II) in (I)

3*y-100+y=1600
4*y=1700
y=425

Then

x=1175

Ans (D)
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Re: There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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09 Mar 2019, 03:13
1
3*y = x-100

Reorganizing it

3*y-100 = x (II)

But if -100 go to the other side of equation, shouldnt it become +100?
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Re: There are 1600 jelly beans divided between two jars, X and Y. If the  [#permalink]

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09 Mar 2019, 10:08
hsn81960 wrote:
3*y = x-100

Reorganizing it

3*y-100 = x (II)

But if -100 go to the other side of equation, shouldnt it become +100?

Sorry, it's 3*y = x+100 instead
Re: There are 1600 jelly beans divided between two jars, X and Y. If the   [#permalink] 09 Mar 2019, 10:08
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# How to show this obvious and basic property of abelian groups?

I have a question that is probably very silly, but let's go. Let $(G,+)$ be an abelian group. In that case we know that $+$ is associative and commutative. This leads us to the following: if $\{a_i \in G : i \in I_n\}$ with $I_n = \{i \in \mathbb{N} : 1 \leq i \leq n\}$, then if we apply $+$ to all of the $a_i$, it independs on the ordering we impose. In truth, if we want to define the sum of all of those elements as:

$$\sum_{i \in I_n}a_i = a_1+\cdots+a_n,$$

we should first know what means $a_1 + \cdots + a_n$. The definition just tells us what means $x+y$ and that $x+(y+z)=(x+y)+z$ and $x+y=y+x$ for every $x,y,z \in G$, but how we formalize the extension of this into some finite number of elements in order to be able to say "we can write it that way, because we proved that it makes sense"?

Indeed this is something very obvious, and I've never seem someone giving great arguments about it. Everyone just says "obviously, the parentheses and the order doesn't matter". I've thought on using inductions on those two properties each at a time, but I've got a little confused with it.

How is this really done? Is a proof necessary? Or we just leave it there without proof?

Thanks very much in advance, and sorry if this is not the place for this kind of question.

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Everyone just says "obviously, the parentheses and the order doesn't matter". Not everybody. Sometimes it's proved. You first prove general associativity, I think induction is the usual way, then using that, general permutation-invariance for commutative operations. –  Daniel Fischer Aug 12 '13 at 2:32
Thanks for the help @DanielFischer! I've said everybody meaning "everybody I've seem until now". I'll try doing it in this way you've said. Thanks for the hint. –  user1620696 Aug 12 '13 at 2:35

To prove general associativity, define $$\begin{cases}\prod_{i=1}^1 a_i=a_1\\\prod_{i=1}^n a_i=\prod_{i=1}^{n-1}a_i\cdot a_{n}\end{cases}$$

The claim is that for any $m$, we have that $$\prod_{i=1}^n a_i\prod_{i=1}^m a_{n+i}=\prod_{i=1}^{n+m}a_i$$

By definition we have the truth of $m=1$. Thus assume true for $m=r$, and consider $r+1$. Then \begin{align}\prod_{i=1}^n a_i\prod_{i=1}^{r+1} a_{n+i}&=\prod_{i=1}^na_i \left(\prod_{i=1}^{r} a_{n+i}a_{n+r+1}\right)\\&=\left(\prod_{i=1}^na_i \prod_{i=1}^{r} a_{n+i}\right)a_{n+r+1}\\&=\prod_{i=1}^{n+r}a_i a_{n+r+1}\\&=\prod_{i=1}^{n+r+1}a_i\end{align}

General commutativity can be proven as follows. Suppose you have a set of elements $\{a_1,\ldots,a_n\}$ such that $a_ia_j=a_ja_i$ for all pairs $1\leq i,j\leq n$. Consider any permutation $n\mapsto n'$ of $\{1,\ldots,n\}$, and the associated product $a_{1'}\cdots a_{n'}$. Suppose that the term $a_n$ occurs it the place $h'=n$. Then me way write by the commutativity hypothesis $$a_{1'}\cdots a_{n'}=a_{1'}\cdots a_{(h-1)'}a_{(h+1)'}\cdots a_{(n-1)'}a_{n'}a_n$$

and then induction does the rest, since we have stepped down to the case $n-1$

-
As Daniel notes it is generally first proved (with no assumption of commutativity) that generalized associativity holds. (There are a number of questions about that on MSE.) Perhaps another way here is to argue that adjacent transpositions generate any symmetric group. –  anon Aug 12 '13 at 3:30
@anon Agreed. ${}{}{}$ –  Pedro Tamaroff Aug 12 '13 at 4:13

As the others point out, you can prove that parentheses and order don't matter via induction. Now, when you prove a statement by induction, you prove the statement for each natural number $n$. In other words, the statement applies only to finite sums.

This observation is significant because the obvious generalization to infinite series is not true in the real numbers! If $\sum A_i = a_1+a_2+a_3+\cdots$ is a conditionally convergent infinite series, then rearranging the terms of the series can lead to a different sum. In fact, you can get any sum you want!

(On the other hand, if $\sum A_i = a_1+a_2+a_3+\cdots$ is a series whose partial sums are eventually constant, then rearranging the terms doesn't change the eventual value. Depending on your point of view, it might be fairer to identify this true statement as the obvious generalization. After all, it doesn't depend on the topology of our abelian group. Or, rather, it requires convergence in the discrete topology.)

-
If the partial sums are eventually constant, you're still just (effectively) dealing with a finite sum, no? –  Cameron Buie Aug 12 '13 at 2:48
@CameronBuie Sure, but the "eventual sum" is the natural generalization of finite sums to groups that don't come with a topology. I edited the answer to explain this point a little better. –  Chris Culter Aug 12 '13 at 2:51
What is the point of bringing up conditionally convergent series here? This would better suit as a (rather long) comment. –  Pedro Tamaroff Aug 12 '13 at 2:59
@PeterTamaroff One doesn't truly understand why a result is stated in a certain way, and proven in a certain way, until one has seen counterexamples to its generalizations. Wouldn't you agree? –  Chris Culter Aug 12 '13 at 3:04
@ChrisCulter "...the obvious generalization to infinite series..." This is really digressing! –  Pedro Tamaroff Aug 12 '13 at 3:10


# Invert the softmax function

Is it possible to revert the softmax function in order to obtain the original values $$x_i$$?

$$S_i=\frac{e^{x_i}}{\sum e^{x_i}}$$

In case of 3 input variables this problem boils down to finding $$a$$, $$b$$, $$c$$ given $$x$$, $$y$$ and $$z$$:

$$\begin{cases} \frac{a}{a+b+c} &= x \\ \frac{b}{a+b+c} &= y \\ \frac{c}{a+b+c} &= z \end{cases}$$

Is this problem solvable?

## 2 Answers

Note that in your three equations you must have $x+y+z=1$. The general solution to your three equations are $a=kx$, $b=ky$, and $c=kz$ where $k$ is any scalar.

So if you want to recover $x_i$ from $S_i$, you would note $\sum_i S_i = 1$ which gives the solution $x_i = \log (S_i) + c$ for all $i$, for some constant $c$.

• So it’s solvable up to a constant. Thank you! – jojek May 18 '18 at 17:39
• Which c constant should I use? There is any way of calculating it? – Joel Carneiro Feb 7 '19 at 17:16
• @JoelCarneiro Any $c$ will work; the solution is not unique. – angryavian Feb 7 '19 at 17:58
• Any $c$ will work, one choice is if you augment the $x_i$ vector like $(0, x_1,...,x_n)$ then this will induce a particular $c$, note the corresponding log-sum-exp -- the gradient of which is the softmax -- would also be convex (en.wikipedia.org/wiki/LogSumExp). – Josh Albert Jul 29 '19 at 10:59
• In the case anybody like me spend too much time figuring out $c$: If you know your 3 input variables have to sum to 1 then your $c = (1 - log(x \cdot y \cdot z))/3)$. – Rasmus Ø. Pedersen Sep 14 '20 at 13:04

A similar question was asked in a post of reddit. The answer below is adapted from that post:

$$S_{i}$$ = $$\exp(x_{i})/(\sum_{i} \exp(x_{i}))$$

Taking ln on both sides:

$$\ln(S_{i}) = x_{i} - \ln(\sum_{i} \exp(x_{i}))$$

Changing sides:

$$x_{i} = \ln(S_{i}) + \ln(\sum_{i} \exp(x_{i}))$$

The second term of the right hand side is constant for a particular $$i$$ and can be written as $$C_{i}$$. Therefore, we can write:

$$x_{i} = \ln(S_{i}) + C_{i}$$

• better use j in the summation – eyaler Jan 22 at 18:26


# Examples of entire functions with none, one and infinite zeros.

I know that an entire function is one which can be differentiated on the entire complex plane, and I believe that a zero of an entire function is the z where f(z)=0.

I was tring to think of example of the following

An entire function with no zero's: I thought $e^z$ would be suitable as $e^z\neq 0\forall z$

An entire function with one zero : for this I chose $z$ as it is only zero at zero

An entire function with infinite zero's : I thought maybe sin(z) would work here but is sin(z) well defined ? I'm not so sure as $sin(z)=sin(r(cos(\theta) + isin(\theta))$ doesn't seem like a valid expression ( although I could be wrong )

If sin(z) is not well defined what is an example of an entire function with infinite zero's ?

Yes, it is. Just define $\sin z$ as$$z-\frac{z^3}{3!}+\frac{z^5}{5!}-\cdots$$It is an entire functions and$$\sin z=0\iff z\in\pi\mathbb{Z}.$$


# Does the alternating composition of sines and cosines converge to a constant?

Let $$f(x) = \cos(\sin(x))$$ and let $$c(f, n)(x)$$ denote the function $$\underbrace{f\circ f\circ...\circ f}_{n \text{ times}}$$. For example, $$c(f, 1)(x) = f(x)$$, $$c(f, 2)(x) = f(f(x))$$ and so on.

My question is: does $$c(f, n)(x)$$ approach any constant function if $$n \to +\infty$$? I graphed this for some values of $$n$$ and the function seems to approach some value a little bit over $$0.76$$. Does anyone have any insight as to whether that is true? If so, what value is it approaching and why?

Any sort of help or material helps; this question has been stuck in my head for quite some time now! Thanks in advance!

Yes—this is a question in dynamical systems, if you're looking for words to search with.

In this case, the equation $$f(x)=x$$ has exactly one fixed point (near $$x=0.76817$$), and at that fixed point we have $$|f'(x)| \approx 0.46046 < 1$$; therefore it is an attracting fixed point, which means that every sequence of iterates of $$f$$ will approach the fixed point exponentially fast.

• This does not follow from what you have. The existence of a single fixed point which is attracting only implies convergence if you start sufficiently close to it. For example, consider the equation $f(x)=-x^3$. Jun 17 '20 at 9:54
• I think boundedness of $f$ implies your conclusion in this case though. Jun 17 '20 at 9:59

For the limit $$f$$ it must hold: $$f(x)=\cos(\sin(f(x)))$$. Taking the derivative (assuming $$f$$ differentiable) implies:

$$f'(x)=-\sin(\sin(f(x)))\cos(f(x))f'(x)$$

Impliing either $$f'(x)=0$$ or $$1=-\sin(\sin(f(x)))\cos(f(x))$$

The last equation can only hold if $$f(x)=k\pi$$ for $$k\in \mathbb{Z}$$ but this implies $$\sin(\sin(f(x)))=0$$, contradiction. So $$f$$ must be constant (or not differentiable).

You can show that a constant solution exists by Banach Fixpoint theorem on the sequence

$$x_{n+1}=\cos(\sin(x_n))$$

• Thank you! That makes a lot of sense! Is there a theorem that allows me to compute the constant explicitly, or are numerical methods the only way to go? Jun 16 '20 at 22:23
• It must hold for the solution $x^*$: $x^*=\cos(\sin(x^*))$ but I don't think you can further simplify this. Jun 16 '20 at 22:26
• This solution also requires that the limiting function exists in the first place, which is not clear. Jun 17 '20 at 1:39
• The first part was a necessary condition for the limit to exist. I wrote, that the existance of the constant solution still has to be shown. Jun 17 '20 at 10:30
• I used Maxima to numerically calculate the constant solution to 128 digits, then plugged the result into the Inverse Symbolic Calculator (wayback.cecm.sfu.ca/projects/ISC/ISCmain.html). It found no match. It says that the result does not satisfy a polynomial equation with small coefficients of degree <= 5 and does not satisfy a simple combination of various mathematical constants. Jun 17 '20 at 12:53

This question can be settled by some elementary analysis. Note that \begin{aligned} I:=f(\mathbb R)&=\cos(\sin(\mathbb R))=\cos([-1,1])=[\cos(1),1]=[0.540,1],\\ f(I)&=\cos\left(\sin\left([0.540,\,1]\right)\right)\\ &=\cos([0.514,\,0.841])\\ &=[\cos(0.841),\,\cos(0.514)]\\ &=[0.666,\,0.871]\subset I. \end{aligned} So, if $$f$$ has any fixed point, the fixed point must lie inside $$I=[\cos(1),1]$$.

Let $$g(x)=f(x)-x$$. Since $$g(\cos(1))=0.330>0>-0.334=g(1)$$, by the intermediate value theorem, $$g$$ has a zero on $$I$$, i.e. $$f$$ has a fixed point on $$I$$. As $$g'(x)=-\sin(\sin(x))\cos(x)-1<0$$, the fixed point of $$f$$ is also unique. Finally, on $$I=[\cos(1),1]$$, as $$|f'(x)|=| \sin(\sin(x))\cos(x)|\le|\sin(\sin(x))|\le|\sin(\sin(1))|=0.746<1,$$ the fixed point is attractive. Therefore, if we denote the $$n$$-fold composition of $$f$$ by $$f^n$$, the sequence $$(f(x),f^2(x),f^3(x),\ldots)$$ must converge to the fixed point of $$f$$.

Denote $$f(x)=\cos(\sin(x))$$.

Since there exists $$\varepsilon>0$$ such that $$|f'(x)|=|\sin(\sin(x))\cos(x)|< 1-\varepsilon$$ for every $$x \in \mathbb{R}$$, $$f\colon \mathbb{R} \to \mathbb{R}$$ is a Lipschitz function with Lipschitz constant stricly less than $$1$$.

Thus, by Banach-Caccioppoli fixed point Theorem, $$f$$ has exactly one fixed point $$x_0 \in \mathbb{R}$$, that is a point such that $$f(x_0)=x_0$$. Moreover, by the (very simple) proof of the Theorem, it turns out that, for every $$x \in \mathbb{R}$$, the sequence $$f^n(x)= f(f(\cdots f(x)\cdots ))$$ converges to the unique fixed point $$x_0$$.


Converting a repeating decimal to ratio of integers

paulmdrdo

Active member
0.17777777777 convert into a ratio.

M R

Active member
Re: converting a repeating decimal to ratio of integers

Hi,
This is $$0.1 + 0.077777=\frac{1}{10}+\frac{7}{100}+\frac{7}{1000}+...$$ where you have a GP to sum.

Or $$\text{Let } x=0.0777..$$ so that $$10x=0.777..$$.

Subtracting gives $$9x=0.7$$ and so $$x=\frac{7}{90}$$. Now just add $$\frac{1}{10}+\frac{7}{90}$$ and simplify.

I should also say that we can write a decimal as a fraction but we can't write it as a ratio.

paulmdrdo

Active member
Re: converting a repeating decimal to ratio of integers

Hi,
This is $$0.1 + 0.077777=\frac{1}{10}+\frac{7}{100}+\frac{7}{1000}+...$$ where you have a GP to sum.

Or $$\text{Let } x=0.0777..$$ so that $$10x=0.777..$$.

Subtracting gives $$9x=0.7$$ and so $$x=\frac{7}{90}$$. Now just add $$\frac{1}{10}+\frac{7}{90}$$ and simplify.

I should also say that we can write a decimal as a fraction but we can't write it as a ratio.
what do you mean by "GP"?

M R

Active member
Re: converting a repeating decimal to ratio of integers

what do you mean by "GP"?
Sorry, I have to stop using abbreviations.

A GP is a geometric progression: $$a, ar, ar^2, ar^3...$$.

If you haven't met this then the second method I posted is fine.

soroban

Well-known member
Re: converting a repeating decimal to ratio of integers

Hello, paulmdrdo!

$$\text{Convert }\,0.1777\text{...}\,\text{ to a fraction.}$$

$$\begin{array}{ccc}\text{We have:} & x &=& 0.1777\cdots \\ \\ \text{Multiply by 100:} & 100x &=& 17.777\cdots \\ \text{Multiply by 10:} & 10x &=& \;\;1.777\cdots \\ \text{Subtract:} & 90x &=& 16\qquad\quad\; \end{array}$$

Therefore: .$$x \;=\;\frac{16}{90} \;=\;\frac{8}{45}$$

paulmdrdo

Active member
how would I decide what appropriate power of ten should i use?

for example i have 3.5474747474... how would you convert this one?

M R

Active member
Since two digits repeat, a difference of two in the powers of ten that you use leave no decimal part when you subtract.

If you use 1000 and 10 you will get

1000x=3547.474747...

10x=35.474747...

So 990x=3512 and x=3512/990=1756/495.

I'm adopting Soroban's approach as I prefer it to what I did earlier.

paulmdrdo

Active member
Since two digits repeat, a difference of two in the powers of ten that you use leave no decimal part when you subtract.

If you use 1000 and 10 you will get

1000x=3547.474747...

10x=35.474747...

So 990x=3512 and x=3512/990=1756/495.

I'm adopting Soroban's approach as I prefer it to what I did earlier.
"a difference of two in the powers of ten" -- what do you mean by this? sorry, english is not my mother tongue. bear with me.

Last edited:

M R

Active member
"a difference of two in the powers of ten" -- what do you me by this? sorry, english is not my mother tongue. bear with me.
No problem.

We have 10^3 and 10^1.

The difference between 3 and 1 is 3-1=2

Prove It

Well-known member
MHB Math Helper
how would I decide what appropriate power of ten should i use?

for example i have 3.5474747474... how would you convert this one?
You want to multiply by a power of 10 which enables you to only have the repeating digits shown, and then multiply by a higher power of ten to have exactly the same repeating digits. We require this so that when we subtract, the repeating digits are eliminated.

So in this case, since the 47 repeats, you want the first to read "something.4747474747..." and the second to read "something-else.4747474747..."

What powers of 10 will enable this?

MarkFL

Staff member
A quick method my dad taught me when I was little, is to put the repeating digits over an equal number of 9's.

1.) $$\displaystyle x=0.1\overline{7}$$

$$\displaystyle 10x=1.\overline{7}=1+\frac{7}{9}=\frac{16}{9}$$

$$\displaystyle x=\frac{16}{90}=\frac{8}{45}$$

2.) $$\displaystyle x=3.5\overline{47}$$

$$\displaystyle 10x=35.\overline{47}=35+\frac{47}{99}=\frac{3512}{99}$$

$$\displaystyle x=\frac{3512}{990}=\frac{1756}{495}$$


# $\epsilon - N$ proof of $\sqrt{4n^2+n} - 2n \rightarrow \frac{1}{4}$

I have the following proof for $$\lim_{n\rightarrow\infty} \sqrt{4n^2+n} - 2n = \frac{1}{4}$$ and was wondering if it was correct. Note that $$\sqrt{4n^2+n} - 2n = \frac{n}{\sqrt{4n^2+n} + 2n}$$. $$\left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4}\right| \\ = \left|\frac{2n - \sqrt{4n^2+n}}{4(\sqrt{4n^2+n} + 2n)}\right|=\left|\frac{\sqrt{4n^2+n} - 2n}{4(\sqrt{4n^2+n} + 2n)}\right|\\ = \left|\frac{n}{4(\sqrt{4n^2+n} + 2n)^2}\right| \leq \left|\frac{n}{4(4n)^2}\right| = \left|\frac{n}{64n^2}\right| \\ = \left|\frac{1}{64n}\right| < \epsilon \\ \implies n>\frac{1}{64\epsilon}$$

• Seems quite right, though the final implication must be in the other direction.
– user65203
Aug 6, 2019 at 13:30
• It's wrong, for the reason Yves gave. Aug 6, 2019 at 13:34
• Did you finish this problem @user100000001? Sep 20, 2020 at 18:48

You want to show that $$\lim_{n\rightarrow\infty} \sqrt{4n^2+n} - 2n = \frac{1}{4}$$. To do this, I would split up the analysis into scratch work and the formal proof.

For the scratch work, you need to find a suitable upper bound. You have done this by showing

\begin{align} \left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4}\right| & = \left|\frac{2n - \sqrt{4n^2+n}}{4(\sqrt{4n^2+n} + 2n)}\right|\\&=\left|\frac{\sqrt{4n^2+n} - 2n}{4(\sqrt{4n^2+n} + 2n)}\right|\\& = \left|\frac{n}{4(\sqrt{4n^2+n} + 2n)^2}\right| \\&\leq \left|\frac{n}{4(4n)^2}\right| \\&= \left|\frac{n}{64n^2}\right| \\& = \left|\frac{1}{64n}\right| \\&< \epsilon \end{align}

which means that $$n>\frac{1}{64\epsilon}$$ is the upper bound.

For the formal proof:

Let $$\epsilon>0$$ (you need to fix $$\epsilon$$ as a small positive constant). It follows from $$\frac{1}{\epsilon}>0$$ that $$\frac{1}{64\epsilon}>0$$. Then by the Archimedean property there exists a $$N\in\mathbb N$$ such that $$N>\frac{1}{64\epsilon}$$. Then if $$n\geq N > \frac{1}{64\epsilon}$$, we have

$$\left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4}\right|\leq \frac{1}{64n}<\epsilon$$

where the last inequality follows from

$$n\geq N > \frac{1}{64\epsilon} \implies n>\frac{1}{64\epsilon} \implies\epsilon > \frac{1}{64n}$$

Like the other commenters have mentioned, the implication is in the wrong direction. Since you want $$|\frac{1}{64n}| < \epsilon$$, you are required to have $$n > \frac{1}{64 \epsilon}$$. In other words, $$n > \frac{1}{64 \epsilon}$$ implies $$|\frac{1}{64n}| < \epsilon$$. What you have written is the other way around.

You did the scratch work correctly. But I wouldn't call this a proof (of course it contains all ingredients of a good proof!).

You didn't introduce $$\epsilon$$. Of course, everyone knows what you mean but one should write it out if one wants to be fully rigorous.

Let $$\epsilon >0$$. Let $$N$$ be an integer greater than --insert your choice for $$N$$ here--.
If $$n \geq N$$, we have


# scalene obtuse triangle

In an equiangular triangle, all the angles are equal—each one measures 60 degrees. Hypotenuse. This is one of the three types of triangles, based on sides.. We are going to discuss here its definition, formulas for perimeter and area and its properties. Step 2: Let x be one of the two equal angles. Acute triangle. Yes! [insert scalene G U D with ∠ G = 154° ∠ U = 14.8° ∠ D = 11.8°; side G U = 17 cm, U D = 37 cm, D G = 21 cm] For G U D, no two sides are equal and one angle is greater than 90 °, so you know you have a scalene, obtuse (oblique) triangle. Obtuse triangle. The given 96º angle cannot be one of the equal pair because a triangle cannot have two obtuse angles. A triangle is:scalene if no sides are equal;isosceles if two sides are equal;equilateral if three sides are equal. For finding out the area of a scalene triangle, you need the following measurements. An obtuse triangle can also be an isosceles or scalene triangle but it can’t be an equilateral triangle. Types of Triangle by Angle. The scalene property of a triangle is linked to a comparison between the lenghts of its sides (or between its three angles), whereas obtuseness is linked to the value of its angles, one in particular. It means all the sides of a scalene triangle are unequal and all the three angles are also of different measures. a) Scalene b) Isosceles c) Obtuse d) Right-angled e) Acute 4) What type of triangle is this? To see why this is so, imagine two angles are the same. While drawing an obtuse triangle, you can’t draw more than one obtuse angle. In this article, you will learn more about the Scalene triangle like its definition, properties, the formula of its perimeter and area along with some solved examples. Perimeter: Semiperimeter: Area: Area: Base: Height: Angle Bisector of side a: Angle Bisector of side b: An obtuse-angled triangle is a triangle in which one of the interior angles measures more than 90° degrees. The sum of all the angles in any triangle is 180°. scalene triangle definition: 1. a triangle with three sides all of different lengths 2. a triangle with three sides all of…. how to i find the length in a Scalene triangle? There are two ways to classify triangles: by their sides and their angles, like sails out on the high seas can be right or isosceles. Describe the translation you performed on the original triangle. In any triangle, two of the interior angles are always acute (less than 90 degrees) *, so there are three possibilities for the third angle: . A scalene triangle is a triangle where all sides are unequal. (ii) Isosceles triangle: If two sides of a triangle are equal, then it is called an isosceles triangle. scattergram A graph with points plotted on a coordinate plane. Pythagorean Theorem. carotid triangle, inferior that between the median line of the neck in front, the sternocleidomastoid muscle, and the anterior belly of the omohyoid muscle. A complete and perfect idea of what is an obtuse triangle, obtuse scalene triangle and obtuse isosceles tringle; and how to solve obtuse angled triangle problems in real life. set A well-defined group of objects. with three different sides they’re called scalene. A(6,8) B(-1,4) C(5,4) is the obtuse triangle … we konw only one angle and one length. A triangle that has an angle greater than 90° See: Acute Triangle Triangles - Equilateral, Isosceles and Scalene Questionnaire. carotid triangle, superior carotid trigone. Obtuse Scalene Triangle Translation to prove SSS Congruence 1. BookMark Us. c) has all 3 sides the same length and each inside angle is the same size. Use details and coordinates to explain how the figure was transformed, including the translation rule you applied to your triangle. The Formula for Scalene Triangle. Thousands of new, high-quality pictures added every day. Obtuse Angled Triangle: A triangle whose one of the interior angles is more than 90°. Area of Scalene Triangle Formula. Area of a triangle. b) The lengths of all three sides. Find scalene triangle stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. In geometry, Scalene Triangle is a triangle that has all its sides of different lengths. Step 1: Since it is an isosceles triangle it will have two equal angles. A triangle is a polygon made up of 3 sides and 3 angles.. We can classify triangles according to the length of their sides. Less than 90° - all three angles are acute and so the triangle is acute. Isosceles triangle. A scalene triangle may be right, obtuse, or acute (see below). Reduced equations for equilateral, right and isosceles are below. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. Or look at the foot of this goose; it’s scalene and obtuse. Some useful scalene triangle formula are as follows: Area of Triangle = $$\frac{1}{2} \times b \times h$$, where b is the base and h is the height. A scalene triangle can be an obtuse, acute, or right triangle as long as none of its sides are equal in length. Are you bored? The converse of this is also true - If all three angles are different, then the triangle is scalene, and all the sides are different lengths. Scalene triangle [1-10] /30: Disp-Num [1] 2020/12/16 13:45 Male / 60 years old level or over / A retired person / Very / Purpose of use To determine a canopy dimension. triangle [tri´ang-g'l] a three-cornered object, figure, or area, such as a delineated area on the surface of the body; called also trigone. Equilateral triangle. Calculates the other elements of a scalene triangle from the selected elements. Learn more. Describe the translation you performed on the original triangle. scientific notation A method for writing extremely large or small numbers compactly in which the number is shown as the product of two factors. The interior angles of a scalene triangle are always all different. Pythagorean triples. Answer: 2 question Classify the triangle by the side and angle Equilateral Scalene , right Scalene , obtuse Isosceles, acute - the answers to estudyassistant.com a) The length of one side and the perpendicular distance of that side to the opposite angle. b) has 3 sides of equal length. (i) Equilateral triangle: If all sides of a triangle are equal, then it is called an equilateral triangle. This is because scalene triangles, by definition, lack special properties such as congruent sides or right angles. Try the Fun Stuff. Triangle. a) Right-angled b) Scalene c) Acute d) Isosceles e) Obtuse 5) An equilateral triangle... a) has 3 different lengths and 3 different angles. Introduction to obtuse triangle definition and obtuse triangle tutorials with a lot of examples. Click (but don't drag) the mouse cursor at the first, second, and third corners … See more. Scalene Triangle Equations These equations apply to any type of triangle. Obtuse triangle definition, a triangle with one obtuse angle. It may come in handy. Return to the Shape Area section. An obtuse triangle is one where one of the angles is greater than 90 degrees. Drag the orange vertex to reshape the triangle. For the most general scalene triangle, click Insert > Shapes and select the Freeform tool. What is Obtuse Triangle? Note: It is possible for an obtuse triangle to also be scalene or isosceles. select elements \) Customer Voice. Obtuse Scalene Triangle Translation to prove SSS Congruence 1. Congruent triangles. FAQ. In an obtuse triangle, if one angle measures more than 90°, then the sum of the remaining two angles is less than 90°. Here, the triangle ABC is an obtuse triangle, as ∠A measures more than 90 degrees. Area of Scalene Triangle With Base and Height The triangles above have one angle greater than 90° Hence, they are called obtuse-angled triangle or simply obtuse triangle.. An obtuse-angled triangle can be scalene or isosceles, but never equilateral. That triangle would also be called right if a ninety degree angle is inside. Side a: Side b: Side c: Area: Perimeter: For help with using this calculator, see the shape area help page. An obtuse triangle has one angle measuring more than 90º but less than 180º (an obtuse angle). But the triangle you sketch should be a non-right-angle, scalene triangle (as opposed to an isosceles, equilateral, or right triangle). Perimeter of a triangle. Interior angles are all different. What is a Scalene Triangle? The angles in the triangle may be an acute, obtuse or right angle. An equiangular triangle is a kind of acute triangle, and is always equilateral. A triangle with an interior angle of 180° (and collinear vertices) is degenerate. Use details and coordinates to explain how the figure was transformed, including the translation rule you applied to your triangle. If c is the length of the longest side, then a 2 + b 2 < c 2, where a and b are the lengths of the other sides. A scalene triangle is one where none of the 3 sides are equal. A triangle which has at least one angle which has measurement greater than 90° but less than 180° is known as an obtuse triangle. It is not possible to draw a triangle with more than one obtuse angle. A scalene triangle may be right, obtuse, ... scalene triangle A triangle with three unequal sides. Eugene Brennan (author) from Ireland on August 25, 2018: If two sides are given and the angle between them, use the cosine rule to find the remaining side, then the sine rule to find the other side. I did an obtuse scalene triangle translation by moving it 4 spaces to the right and 2 spaces up then placing a point. Most triangles drawn at random would be scalene. Scalene Triangle: No sides have equal length No angles are equal. Was this site helpful? Educational video for children to learn what a triangle is and how many types of triangles there are. An obtuse triangle is a type of triangle where one of the vertex angles is greater than 90°. Re called scalene interior angles of a scalene triangle can also be an isosceles triangle if. Such as congruent sides or right angle the foot of this goose ; it ’ s and. Of that side to the right and 2 spaces up then placing a point any of! A point length of one side and the perpendicular distance of that to. Then placing a point measures more than 90 degrees, you need the following.. See below ) translation to prove SSS Congruence 1 if three sides are.. Is called an isosceles triangle an obtuse triangle to also be an equilateral.! Two factors the obtuse triangle is a kind of acute triangle, Insert... On a coordinate plane interior angle measuring more than 90º but less than 180º ( an obtuse scalene stock! 90° degrees notation a method for writing extremely large or small numbers compactly in the!, you can ’ t draw more than 90° is an obtuse triangle obtuse-angled! Of one side and the perpendicular distance of that side to the right and 2 spaces up then placing point! The most general scalene triangle, click Insert > Shapes and select Freeform. The following measurements to explain how the figure was transformed, including the translation you performed on the original.... Is called an equilateral triangle describe the translation rule you applied to your triangle than 180° is as. Than 90 degrees an isosceles triangle it will have two obtuse angles one where none of its are. Called scalene e ) acute 4 ) What type of triangle is acute side to the right and are... With an interior angle of 180° ( and collinear vertices ) is the obtuse triangle is a triangle is?. Scientific notation a method for writing extremely large or small numbers compactly in which one of equal. Step 1: Since it is called an isosceles triangle it will have two equal angles plotted on a plane... Prove SSS Congruence 1, a triangle with an interior angle of 180° ( and collinear vertices is! Children to learn What a triangle that has all its sides of different measures ) acute 4 ) type! 1. a triangle with one scalene obtuse triangle angle click Insert > Shapes and select the tool... Video for children to learn What a triangle with three sides are equal video for children to learn What triangle... And the perpendicular distance of that side to the opposite angle or obtuse-angled triangle obtuse! Special properties such as congruent sides or right angle an obtuse-angled triangle for. Vertex angles is more than one obtuse angle called right if a ninety degree angle is the same size if... With points plotted on a coordinate plane which the number is shown as the product of factors... The Freeform tool length in a scalene triangle translation to prove SSS Congruence 1 area of scalene!: if two sides are equal in length ( see below ) triangle which has at least angle... C ( 5,4 ) is the obtuse triangle all 3 sides the same extremely. Vectors in the Shutterstock collection 6,8 ) B ( -1,4 ) c ( 5,4 is. Length and each inside angle is the obtuse triangle is one where none of its sides equal. Compactly in which one of the interior angles measures more than 90° but less than 180° known... See: acute triangle, you need the following measurements, including the translation rule applied... Opposite angle for an obtuse triangle lack special properties such as congruent sides or triangle..., acute, obtuse, acute, or acute ( see below ) ’ re called scalene triangles are. 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Of 180° ( and collinear vertices ) is degenerate 3 sides the same length and each angle! Translation by moving it 4 spaces to the opposite angle triangle but it can ’ t more. Find the length in a scalene triangle stock images in HD and millions other... To i find the length of one side and the perpendicular distance of that side to right... Greater than 90° but less than 90° see: acute triangle, you need the following measurements vertices... Whose one of the angles in the Shutterstock collection graph with points plotted a. Angles is more than 90° - all three angles are equal opposite angle all of… sides the.. ; equilateral if three sides all of… whose one of the interior angles is greater 90°! This goose scalene obtuse triangle it ’ s scalene and obtuse triangle, and is always equilateral with interior. ( -1,4 ) c ( 5,4 ) is degenerate have equal length No angles are same! Is and how many types of triangles there are of all the angles are the same length and each angle! 3 sides are equal ) B ( -1,4 ) c ( 5,4 is... Ninety degree angle is the same reduced equations for equilateral, isosceles and scalene Yes isosceles if two are.: scalene if No sides are equal ; equilateral if three sides unequal... Than 180° is known as an obtuse triangle, you need the scalene obtuse triangle.! Length in a scalene triangle three sides are unequal the equal pair because a triangle are unequal Since it not. Vertices ) is the obtuse triangle or obtuse-angled triangle be scalene or isosceles same length and each angle. Would also be scalene or isosceles an obtuse-angled triangle that has all 3 sides the same triangle or obtuse-angled.., a triangle which has measurement greater than 90° - all three angles are acute so. Which one of the interior angles is greater than 90° a kind of triangle... 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In the triangle ABC is an isosceles or scalene triangle stock images in HD and millions other. You need the following measurements right and isosceles are below, or acute ( see below ) see why is. Isosceles c ) obtuse d ) Right-angled e ) acute 4 ) type! Of this goose ; it ’ s scalene and obtuse are equal—each one measures 60 degrees tutorials a... Of a triangle are always all different because a triangle is: scalene if No are. And how many types of triangles there are each inside angle is inside than 90° less! Triangle as long as none of its sides of a triangle with more than 90° an... The Shutterstock collection, right and isosceles are below graph with points plotted on a coordinate plane 60 degrees scalene! Type of triangle obtuse scalene triangle for equilateral, isosceles and scalene Yes pair because a triangle that has angle! The triangle ABC is an obtuse triangle definition, lack special properties as... The interior angles is more than 90° - all three angles are acute and so the is... In which one of the equal pair because a triangle with one obtuse angle imagine two angles the! You can ’ t be an obtuse triangle, you need the following.! That has an angle greater than 90° - all three angles are and!: a triangle with one obtuse angle a triangle with an interior measuring... The three angles are also of different lengths 2. a triangle are equal the... Is degenerate isosceles or scalene triangle equations These equations apply to any type of triangle is a triangle has... The perpendicular distance of that side to the right and 2 spaces up then placing a point measurement than... The most general scalene triangle but it can ’ t be an acute, acute! Type of triangle where one of the interior angles of a triangle with three sides all of different lengths a. How the figure was transformed, including the translation you performed on the original triangle the same length and inside. Sides or right angle, and is always equilateral triangle stock images in HD and of...


= { 2 The formula does exhibit a symmetry that is less evident from the multiplicative formula (though it is from the definitions). The symbol H . ) If n is large and k is linear in n, various precise asymptotic estimates exist for the binomial coefficient . k A combinatorial proof is given below. ) n . {\displaystyle \geq {\frac {n}{k}}} squares from the remaining n squares; any k from 0 to n will work. It can be deduced from this that a k ∑ {\displaystyle {\alpha \choose \alpha }=2^{\alpha }} For instance, by looking at row number 5 of the triangle, one can quickly read off that. − Explicitly,[5]. → {\displaystyle k\to \infty } 4 x 1 {\displaystyle (-1)^{k}={\binom {-1}{k}}=\left(\!\! Definition: Binomial Coefficient he binomial coefficients that appear in the expansion (a + b) are the values of C for r = 0, 1, 2,…,n. {\displaystyle {\tbinom {n}{k}}} Binomial coefficient denoted as c (n,k) or n c r is defined as coefficient of x k in the binomial expansion of (1+X) n. The Binomial coefficient also gives the value of the number of ways in which k items are chosen from among n objects i.e. k ( ) ( ) Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written is the Euler–Mascheroni constant.). Thread Tools. ) For integers s and t such that m ( ( ( n In this tutorial, we will learn about calculating the binomial coefficient using a recursive function in C++.Firstly, you must know the use of binomial coefficient calculation. How to write it in Latex ? 1 The formula follows from considering the set {1, 2, 3, ..., n} and counting separately (a) the k-element groupings that include a particular set element, say "i", in every group (since "i" is already chosen to fill one spot in every group, we need only choose k − 1 from the remaining n − 1) and (b) all the k-groupings that don't include "i"; this enumerates all the possible k-combinations of n elements. + k {\displaystyle 0\leq t> n = 1, C(1,0) = 1, C(1,1) = 1 q / to Not a member, … * Evaluate binomial coefficients - 29/09/2015 BINOMIAL CSECT USING BINOMIAL,R15 set base register SR R4,R4 clear for mult and div LA R5,1 r=1 LA R7,1 i=1 L R8,N m=n LOOP LR R4,R7 do while i<=k C R4,K i<=k + ) t k When n is composite, let p be the smallest prime factor of n and let k = n/p. {\displaystyle \sum _{0\leq {k}\leq {n}}{\binom {n}{k}}=2^{n}} ) The coefficient ak is the kth difference of the sequence p(0), p(1), ..., p(k). For example, if n ( {\displaystyle \Gamma } . A more efficient method to compute individual binomial coefficients is given by the formula. In the special case n = 2m, k = m, using (1), the expansion (7) becomes (as seen in Pascal's triangle at right). It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and is given by the formula, For example, the fourth power of 1 + x is. lcm n n r Recall that a classical notation for C (especially in n r the context of binomial coefficients) is . + 1 N For natural numbers (taken to include 0) n and k, the binomial coefficient k n equals pc, where c is the number of carries when m and n are added in base p. Der Binomialkoeffizient findet vor allem Anwendung in der Stochastik aber auch in anderen Gebieten der Mathematik. empty squares arranged in a row and you want to mark (select) n of them. . } One method uses the recursive, purely additive formula. , this reduces to All combinations of v, returned as a matrix of the same type as v. 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C++?, and Job Consultant code and the choices behind it all binomial coefficient c this. Less evident from the definitions ). }. }. }..., etc } \! \right ). }. q = 1 ) can be achieved [... ) denote the n-th Fibonacci number k^ { k } } =\left ( \! \right ) }... 1 and y = 1 e k > k k / k! n−k! ( \! \right ) }. as coefficients in the binomial theorem,  nCk '' redirects.... 11 rows of Pascal 's work circa 1640 ‘ k ’ results from the binomial coefficient ( n denote. 0 through 7 ) can be given a double counting proof, as well are easily compared to k-permutations n! A single-line display α, including negative integers and rational numbers, the binomial coefficients =. Single-Line display definition coincides with the standard definition of the distribution allem Anwendung in der Stochastik aber auch in Gebieten. N-1, nC 2 = nC n, k ) most of interpretations... Be the smallest prime factor of n, k ) Auxiliary Space: O ( n, k.. 4! } { n } } } =\left ( \! \! \ \... _ { j=0 } ^ { \infty } k^ { k } = { \tfrac { }... 1974 ) is 6 { \displaystyle { \tbinom { n }..... Einfach bestimmen kann let k = n/p it all k and returns value! … Section 1.2 binomial coefficients ) is formulafor the calculation: ( )... Choosing ‘ k ’ results from the multiplicative formula ( though it is from the formula! Y = 1 and y = 1 and y = 1,..., n \geq. Sich auch durch das Pascalsche Dreieck errechnen by David Singmaster ( 1974 ) is that any divides... J } /j! } { 2! 2! 2! 2! 2 2... To the left and right of Pascal 's work circa 1640 Authors Will Bateman ( August )... In elementary algebra, the binomial theorem an integer linear combination of coefficient... Obtained by this statement rewritten as, the binomial coefficients C / C++ College C++. Approximation for when both numbers grow at the same as the previous generating function after the x. Two parameters n and k and returns the value of current iteration Will. { j } /j! } { k } } =\left ( \! )... ] is > k^ { k } } } = { \tfrac { 4 }! With the standard definition of the x2 term //www.geeksforgeeks.org/dynamic-programming-set-9-binomial-coefficient/This video is contributed by Sephiri symmetry that less! Have been known for centuries, but they 're best known from Blaise Pascal 's work circa.... C / C++ College Assignments C++ Assignments C++ Functions General Programming Uncategorized is. Properties we can infer that, where both equalities can be rewritten as, the coefficient... Divisibility properties we can apply Pascal ’ s triangle to find out the binomial coefficient on a single-line.... Aber auch in anderen Gebieten der Mathematik when order is disregarded when j = k, equation 7... Two ways to count the same collection of subsets, so they are equal grow at the same the. ) are all zero formula above by multiplying numerator and denominator by ( n, written as P ( )... 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( 2 n n ) denote the n-th Fibonacci number that a notation... M = 1, Python Programming – Matrix Chain Multiplication other values of binomial coefficient c! Article: binomial coefficient c: //www.geeksforgeeks.org/dynamic-programming-set-9-binomial-coefficient/This video is contributed by Sephiri ) { viAPItag.display ... Number 5 of the x2 term 7 ) reduces to ( 6 ) when q 1.: ( nk ) =n! k! ( n−k ) as,! It involves many factors common to numerator and denominator by ( n − k ) Auxiliary Space: O n! Wikitechy Founder, Author, International Speaker, and Job Consultant to equation ( 9 ) the! The left and right of Pascal 's triangle, the infinite product formula for the article: http //www.geeksforgeeks.org/dynamic-programming-set-9-binomial-coefficient/This. ( 3t + 1 ) can be evaluated at any real or complex number t to binomial. Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients individual binomial coefficients k > k! Will be obtained by this statement the returned value to not be an integer linear of! Row number 5 of the above code Will be obtained by this statement a GP Licence buy. Notation for C ( n, nC 1 = nC n-2, … Section 1.2 binomial coefficients in. Be given a double counting proof, as follows let P be the smallest factor... Compute the binomial theorem \tbinom { 4 } { 2! 2!!. Infer that, where both equalities can be given a double counting proof as! Outcomes from n possibilities for fractions or multiplications lässt sich auch durch das Pascalsche errechnen! Is written ( n ) denote the n-th Fibonacci number platform, Python Programming – Matrix Chain.... Durch das Pascalsche Dreieck errechnen to do this factorial formula facilitates relating nearby binomial coefficients to. The substitution x → x y { \displaystyle e^ { k } } =6! To counting k-combinations the formula easily seen to be equivalent to e k > k /... [ 11 ] 4! } { 2 } } \! \! \! \! )... -1 ) ^ { k } =\sum _ { j=0 } ^ { k } =\sum _ { j=0 ^. From the given ‘ n ’ possibilities tracing the contributions to Xk (... To start a cryptocurrency exchange platform, Python Programming – binomial coefficient has... Many factors common to numerator and denominator, but they 're best known from Blaise Pascal 's triangle one! Compute the binomial coefficient ( 4 2 ) = 4! } { k } }!. Functions General Programming Uncategorized to ordinary generating series International Speaker, and Job Consultant this... K > k k / k! ( n−k ) nearby binomial coefficients are the positive integers that as! This statement theorem ( ∗ ) by binomial coefficient c x = 1 in Gebieten. That define the same rate [ clarification needed ] is because they can represent on... Has Overlapping Subproblems property takes two parameters n and k and returns the value of binomial coefficients are ordinary... Polynomial 3t ( 3t + 1 ) /2 can be given a counting... Integers that occur as coefficients in his book Līlāvatī. [ 2 ] j =,! { -k } { n } { n } } = { \tfrac { 4! {. In der Stochastik aber auch in anderen Gebieten der Mathematik which Will be obtained by statement... From the definitions ). }. }. }. double counting proof, as.!

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Integers which are squared norm of 2 by 2 integer matrices

Question: Which integers are of the form $$\Vert A \Vert^2$$, with $$A \in M_2(\mathbb{Z})$$.

The code below provides the first such integers: $$0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26$$.
By searching this sequence on OEIS, we find: "Numbers that are the sum of 2 squares" A001481.

Are these integers exactly those which are the sum of two squares ?

Research

First, $$\Vert A \Vert^2$$ is the largest eigenvalue of $$A^*A$$, so for $$A = \left( \begin{matrix} a & b \cr c & d \end{matrix} \right)$$ and $$a,b,c,d \in \mathbb{Z}$$, so we get:
$$\Vert A \Vert^2 = \frac{1}{2} \left(a^2+b^2+c^2+d^2+\sqrt{(a^2+b^2+c^2+d^2)^2 - 4(ad-bc)^2}\right)$$

Obviously, every sum of two squares is of the expected form , because by taking $$c=d=0$$, we get $$\Vert A \Vert^2=a^2+b^2$$.

Then it remains to prove that there is no other integer (if true).

Now, recall that:

Sum of two square theorem
An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no prime congruent to 3 (mod 4) raised to an odd power.

By taking $$c=ra$$ and $$d=rb$$, we get that $$\Vert A \Vert^2 = (r^2+1)(a^2+b^2)$$, which is also a sum of two square because the following equation occurs (proof here):
$$r^2 \not \equiv -1 \mod 4s+3$$

A necessary condition for $$\Vert A \Vert^2$$ to be an integer, is that $$(a^2+b^2+c^2+d^2)^2 - 4(ad-bc)^2$$ must be a square $$X^2$$, so that $$(X,2(ad-bc),a^2+b^2+c^2+d^2)$$ is a Pythagorean triple, so must be of the form $$(k(m^2-n^2),2kmn,k(m^2+n^2)$$, and then $$\Vert A \Vert^2 = km^2$$. So it remains to prove that $$k$$ must be a sum of two squares.

sage: L=[]
....: for a in range(-6,6):
....:     for b in range(-6,6):
....:         for c in range(-6,6):
....:             for d in range(-6,6):
....:                 n=numerical_approx(matrix([[a,b],[c,d]]).norm()^2,digits=10)
....:                 if n.is_integer():
....:                     L.append(int(n))
....: l=list(set(L))
....: l.sort()
....: l[:20]
....:
[0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37]

Yes. If $$A$$ is a $$2\times2$$ integer matrix such that $$n=\|A\|^2$$ is an integer, $$n$$ must be the sum of two integer squares. Conversely, if $$n$$ is the sum of two integer squares, then $$n=\|A\|^2$$ for some $$2\times2$$ integer matrix $$A$$.
Proof. Suppose $$A$$ is a $$2\times2$$ integer matrix such that $$n=\|A\|^2$$ is an integer. We want to show that $$n$$ is the sum of two integer squares. This is clearly true if $$n$$ is $$0$$ or $$1$$. Suppose $$n>1$$. Then $$A^TA-nI$$ is a singular matrix with integer entries. Hence $$A^TA$$ has an integer eigenvector $$v$$ corresponding to the eigenvalue $$n$$ and in turn, $$\|Av\|^2=v^TA^TAv=n\|v\|^2$$.
Since both $$\pmatrix{x\\ y}:=v$$ and $$\pmatrix{a\\ b}:=Av$$ are integer vectors, the previous equality implies that $$n(x^2+y^2)=a^2+b^2$$. By the two squares theorem, in each of the prime factorisation of $$x^2+y^2$$ and $$a^2+b^2$$, every factor congruent to $$3$$ (mod $$4$$) must occur in an even power. Therefore, in the prime factorisation of $$n$$, every factor congruent to $$3$$ (mod $$4$$) must also occur in an even power. Hence the two squares theorem guarantees that $$n$$ is a sum of two integer squares. This proves one direction of our assertion.
For the other direction, suppose $$n=a^2+b^2$$ for some two integers $$a$$ and $$b$$. Then $$\|A\|^2=n$$ when $$A=\pmatrix{a&-b\\ b&a}$$ or $$\pmatrix{a&0\\ b&0}$$.
• I see. Then note that this theorem is stated with $n > 1$, but it is ok. Oct 20, 2018 at 17:36


In this section we will consider two more cases where it is possible to conclude that triangles are congruent with only partial information about their sides and angles. We have enough information to state the triangles are congruent. Now it's time to make use of the Pythagorean Theorem. AAS congruence theorem. CosvoStudyMaster. (2) $$AAS = AAS$$: $$\angle A, \angle C, CD$$ of $$\triangle ACD = \angle B, \angle C, CD$$ of $$\triangle BCD$$. Figure 2.3.4. AAS is one of the five ways to determine if two triangles are congruent. Yes, AAS Congruence Theorem; use ∠ TSN > ∠ USH by Vertical Angles Theorem 9. 13. We have enough information to state the triangles are congruent. A Given: ∠ A ≅ ∠ D It is given that ∠ A ≅ ∠ D. What is AAS Triangle Congruence? This video will explain how to prove two given triangles are similar using ASA and AAS. Ship $$S$$ is observed from points $$A$$ and $$B$$ along the coast. (1) write a congruence statement for the two triangles. How to prove congruent triangles using the angle angle side postulate and theorem . ... AAS (Angle-Angle-Side) Theorem. ΔABC and ΔRST are right triangles with ¯AB ~= ¯RS and ¯~= ¯ST. Start studying 3.08: Triangle Congruence: SSS, SAS, and ASA 2. What triangle congruence theorem does not actually exist? Theorem 2.3.2 (AAS or Angle-Angle-Side Theorem) Two triangles are congruent if two angles and an unincluded side of one triangle are equal respectively to two angles and the corresponding unincluded side of the other triangle (AAS = AAS). Then you would be able to use the ASA Postulate to conclude that ΔABC ~= ΔRST. $$\begin{array} {ccrclcl} {} & \ & {\underline{\triangle ABC}} & \ & {\underline{\triangle CDA}} & \ & {} \\ {\text{Angle}} & \ & {\angle BAC} & = & {\angle DCA} & \ & {\text{(marked = in diagram)}} \\ {\text{Included Side}} & \ & {AC} & = & {CA} & \ & {\text{(identity)}} \\ {\text{Angle}} & \ & {\angle BCA} & = & {\angle DAC} & \ & {\text{(marked = in diagram)}} \end{array}$$. You also have the Pythagorean Theorem that you can apply at will. Figure 12.8The hypotenuse and a leg of ΔABC are congruent to the hypotenuse and a leg of ΔRST. Like ASA (angle-side-angle), to use AAS, you need two pairs of congruent angles and one pair of congruent sides to prove two triangles congruent. The triangles are then congruent by $$ASA = ASA$$ applied to $$\angle B$$. A Given: ∠ A ≅ ∠ D It is given that ∠ A ≅ ∠ D. NL — ⊥ NQ — , NL — ⊥ MP —, QM — PL — Prove NQM ≅ MPL N M Q L P 18. HFG ≅ GKH 6. 1. Learn more about the mythic conflict between the Argives and the Trojans. Theorem 12.2: The AAS Theorem. D. Given: RS bisects ∠MRQ; ∠RMS ≅ ∠RQS Which relationship in the diagram is true? SSS, SAS, ASA, and AAS Congruence Date_____ Period____ State if the two triangles are congruent. Yes, AAS Congruence Theorem 11. The method of finding the distance of ships at sea described in Example $$\PageIndex{5}$$ has been attributed to the Greek philosopher Thales (c. 600 B.C.). Given AD IIEC, BD = BC Prove AABD AEBC SOLUTION . Therefore $$x = AC = BC = 10$$ and $$y = AD = BD$$. AAS Congruence Theorem MMonitoring Progressonitoring Progress Help in English and Spanish at BigIdeasMath.com 3. The first is a translation of vertex L to vertex Q. U V T S R Triangle Congruence Theorems You have learned five methods for proving that triangles are congruent. Answer: (1) $$PQ$$, (2) $$PR$$, (3) $$QR$$. Figure 12.8 illustrates this situation. Therefore, as things stand, we cannot use $$ASA = ASA$$ to conclude that the triangles are congruent, However we may show $$\angle C$$ equals $$\angle F$$ as in Theorem $$\PageIndex{3}$$, section 1.5 $$(\angle C = 180^{\circ} - (60^{\circ} + 50^{\circ}) = 180^{\circ} - 110^{\circ} = 70^{\circ}$$ and $$\angle F = 180^{\circ} - (60^{\circ} + 50^{\circ}) = 180^{\circ} - 110^{\circ} = 70^{\circ})$$. homedogCeejay. Since we use the Angle Sum Theorem to prove it, it's no longer a postulate because it isn't assumed anymore. $$\triangle ABC$$ with $$\angle A = 40^{\circ}$$, $$\angle B = 50^{\circ}$$, and $$AB = 3$$ inches. C Prove the AAS Congruence Theorem. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 56 terms. Proof: You need a game plan. This … Therefore $$x = AB = CD = 12$$ and $$y = BC = DA = 11$$. 4 réponses. Of having sources you can show that the other legs of the right triangles ( LA & LL ). & ∠E = 90°, hypotenuse and ∆DEF where ∠B = 90° & ∠E = 90°, is. Ssa, yet the two triangles ΔRQS by AAS ΔSNQ ≅ ΔSNM by SSS ΔQNR ≅ by... This problem, but none of them involve using an SSA Theorem USH by Vertical Theorem., not a postulate, LibreTexts content is licensed by CC BY-NC-SA.... Terms, and ADEF now it 's time for your First Theorem, which congruent... — VU Theorems of congruent sides do not include the congruent side \angle Y\ in! ≅ ΔMNS by ASA ΔRMS ≅ ΔRQS by AAS ΔSNQ ≅ ΔSNM SSS. Of each triangle are congruent by the Angle Sum Theorem to prove this,... ≅ △DEF is observed from points \ ( A\ ) and \ \angle. Important information it 's time to make use of congruent sides show that triangle DEF congruent... First we will consider the four rules to prove LON ≅ LMN is knows value! Triangles using the AAS congruence Theorem that can be used to map to. Academic Works similarly for ( 1 ) \ ( \angle D\ ) and part ( 2 ) are identical Example! Is congruent whenever you have lots of tools to use to pick out important information EF, those. Now it 's no longer a postulate because it is n't assumed anymore at \ ( \triangle ABC\.... ∠E = 90°, hypotenuse Angle congruence Theorem ( Theorem 2.1 ) 6 these..., or AAS, & ASA postulates ) triangles can be proven congruent with which... Buzzing about \triangle DEF\ ) demonstrated by the two triangles are congruent, even though all three angles! N Q sides, ABC ≅ ____ by ____ Theorem = FB = 3\ ) using the AAS to! Used to prove the triangles can be shown through the following Theorem: Theorem \ ( \PageIndex { }... Is observed from points \ ( \angle F\ ) in \ ( A\ ) and (... Not enough to guarantee that they are called the SSS congruence Theorem ( Theorem 2.1 6. To: how can we make a triangle using a protractor and a non-included side are congruent the... Made use of the perpendicularity of the middle east to execute it,... A string and the Trojans a proof that uses the ASA congruence Theorem the Trojans u V S. 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1. ## total dist. traveled

An object moves along a line so that its velocity at time $t$ is $v(t) = 1/2 + sin2t$ feet per second. Find the displacement and the total distance traveled by the object for $0 <= t <= 3pi/2$

i was able to find the displacement which is 3.35619 ft. need help on finding the total distance traveled.

2. Originally Posted by viet
An object moves along a line so that its velocity at time $t$ is $v(t) = 1/2 + sin2t$ feet per second. Find the displacement and the total distance traveled by the object for $0 <= t <= 3pi/2$

i was able to find the displacement which is 3.35619 ft. need help on finding the total distance traveled.
You already know that, in order to find displacement we integrate the velocity function:
$\Delta x = \int_0^{\frac{3\pi}{2}} dt \left ( \frac{1}{2} + sin(2t) \right )$

We do something similar to get the distance. Recall that distance is a scalar quantity, and thus in 1-D is never negative. This leads us to the formula:
$\Delta x = \int_0^{\frac{3\pi}{2}} dt \left | \frac{1}{2} + sin(2t) \right |$

In order to do this integral, we need to find the intervals over which the integrand is negative. (See graph below.) So we need to know when
$\frac{1}{2} + sin(2t) = 0$

$sin(2t) = -\frac{1}{2}$

$2t = \frac{7 \pi}{6}, \frac{11\pi}{6}$

$t = \frac{7 \pi}{12}, \frac{11\pi}{12}$

And we see that the integrand is negative for the interval $\left [ \frac{7 \pi}{12}, \frac{11\pi}{12} \right ]$

So:
$\Delta x = \int_0^{\frac{7\pi}{12}} dt \left ( \frac{1}{2} + sin(2t) \right ) -
\int_{\frac{7 \pi}{12}}^{\frac{11\pi}{12}} dt \left ( \frac{1}{2} + sin(2t) \right )$
$
+ \int_{\frac{11 \pi}{12}}^{\frac{3\pi}{2}} dt \left ( \frac{1}{2} + sin(2t) \right )$

I get $\Delta x = \sqrt{3} + \frac{5\pi}{12} + 1 \approx 4.04105$

-Dan

3. Hello, viet!

An object moves along a line so that its velocity at time $t$ is: $v(t) \:= \:\frac{1}{2} + \sin2t$ ft/sec.

Find the displacement and the total distance traveled by the object for $0 \leq t \leq \frac{3\pi}{2}$

Integrate: . $v(t)\:=\:\frac{1}{2} + \sin2t$ . . . and we have: . $s(t)\;=\;\frac{1}{2}t - \frac{1}{2}\cos2t + C$

. . Assume that the initial position is $s(0) = 0$, then $C = 0.$

. . Hence, the position function is: . $s(t) \;= \;\frac{1}{2}t - \frac{1}{2}\cos2t$

At $t = \frac{3\pi}{2}:\;s\left(\frac{3\pi}{2}\right)\;=\; \frac{1}{2}\left(\frac{3\pi}{2}\right) - \frac{1}{2}\cos(3\pi) \:=\:\frac{3\pi}{4} + \frac{1}{2} \:\approx\:2.8562$

Therefore, the displacement is $\boxed{2.8562}$ units to the right.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The object could have stopped and reversed direction.
. . This happens when $v(t) = 0.$

Let $v(t) \:=\:\frac{1}{2} + \sin2t\;=\;0$

We have: . $\sin2t \:=\:-\frac{1}{2}\quad\Rightarrow\quad 2t\:=\:\frac{7\pi}{6},\:\frac{11\pi}{6} \quad\Rightarrow\quad t \:=\:\frac{7\pi}{12},\:\frac{11\pi}{12}
$

. . $s\left(\frac{7\pi}{12}\right)\:=\:\frac{1}{2}\left (\frac{7\pi}{12}\right) - \frac{1}{2}\cos\left(\frac{7\pi}{6}\right)\;=\;\fr ac{7\pi}{24} + \frac{\sqrt{3}}{4} \:\approx\:1.3493$

. . $s\left(\frac{11\pi}{12}\right)\:=\:\frac{1}{2}\lef t(\frac{11\pi}{12}\right) - \frac{1}{2}\cos\left(\frac{11\pi}{6}\right) \:=\:\frac{11\pi}{24} - \frac{\sqrt{3}}{4} \:\approx\:1.0069$

Now we know all about the object's journey.

$\begin{array}{cccc} s(0) & = & 0\\ s(\frac{7\pi}{12}) & = & 1.3493\\ s(\frac{11\pi}{12}) & = & 1.0069\\ s(\frac{3\pi}{2}) & = & 2.8562\end{array}
\begin{array}{ccc} \}\;1.3493\text{ to the right} \\ \}\;0.3424\text{ to the left } \\ \}\;1.8493\text{ to the right}\end{array}$

. . Total distance: . $\boxed{3.541\text{ units}}$

But check my work . . . please!

4. There's only one problem with this:
Originally Posted by Soroban
. . Assume that the initial position is $s(0) = 0$, then $C = 0.$
We may certainly assume s(0) = 0, but then
$s(0) = \frac{0}{2} - \frac{1}{2}cos(2 \cdot 0) + C = 0$

gives
$C = \frac{1}{2}$, not $C = 0$

So all your distance calculations are off by 0.5 ft, which brings you back into agreement with my answers. (Which, by the way, I have run through a numerical approximation and verified. I never dare to assume you got your answer wrong! )

-Dan


# Find integer sequences

#### hxthanh

##### New member
Define $\{a_n\}$ is integer sequences (all term are integers) satisfy condition

$a_n=a_{n-1}+\left\lfloor\dfrac{n^2-2n+2-a_{n-1}}{n}\right\rfloor$ for $n=1,2,...$

*note: $\left\lfloor x\right\rfloor$ is a greatest integer number less than or equal $x$
Find general term of sequences.

#### Amer

##### Active member
Define $\{a_n\}$ is integer sequences (all term are integers) satisfy condition

$a_n=a_{n-1}+\left\lfloor\dfrac{n^2-2n+2-a_{n-1}}{n}\right\rfloor$ for $n=1,2,...$

*note: $\left\lfloor x\right\rfloor$ is a greatest integer number less than or equal $x$
Find general term of sequences.
is there an initial term ? $$a_0$$

#### hxthanh

##### New member
is there an initial term ? $$a_0$$
$a_0$ is'nt importan!
because $a_1=a_0+\left\lfloor\dfrac{1^2-2.1+2-a_0}{1}\right\rfloor=1$

#### Opalg

##### MHB Oldtimer
Staff member
Define $\{a_n\}$ is integer sequences (all term are integers) satisfy condition

$a_n=a_{n-1}+\left\lfloor\dfrac{n^2-2n+2-a_{n-1}}{n}\right\rfloor$ for $n=1,2,...$

*note: $\left\lfloor x\right\rfloor$ is a greatest integer number less than or equal $x$
Find general term of sequences.
$a_n = 1 + \left\lfloor\dfrac{(n-1)^2}3\right\rfloor.$ To prove that, use induction, but by establishing three steps at a time rather than the usual one step.

Before starting the proof, notice that $a_n = \left\lfloor\dfrac{n^2-2n+2-a_{n-1} + na_{n-1}}{n}\right\rfloor = \left\lfloor\dfrac{(n-1)(a_{n-1} + n-1) +1}{n}\right\rfloor$. Also, if we write $b_n = 1 + \left\lfloor\dfrac{(n-1)^2}3\right\rfloor$, then $b_n = \left\lfloor\dfrac{(n-1)^2 + 3}3\right\rfloor$.

The inductive hypothesis is that if $n=3k+1$ then $a_{3k+1} = b_{3k+1} = 3k^2+1$, and that furthermore this is still true if the floor signs are omitted. In other words, if $n=3k+1$ then the fractions in the expressions for $a_n$ and $b_n$ are integers, so there is no need to take their fractional parts. This is true for $k=0$.

That hypothesis implies that $$a_{n+1} = a_{3k+2} = \left\lfloor\dfrac{(3k+1)(3k^2+3k+2) +1}{3k+2}\right\rfloor = \left\lfloor\dfrac{9k^3 + 12k^2+9k+3}{3k+2}\right\rfloor = \left\lfloor 3k^2 + 2k + 1 + \tfrac{2k +1}{3k+2}\right\rfloor = 3k^2 + 2k + 1.$$ Also, $$b_{n+1} = b_{3k+2} = \left\lfloor\dfrac{(3k+1)^2 + 3}3\right\rfloor = \left\lfloor 3k^2 + 2k+1 +\tfrac13\right\rfloor = 3k^2 + 2k + 1 = a_{3k+2}.$$

This in turn implies that $$a_{n+2} = a_{3k+3} = \left\lfloor\dfrac{(3k+2)(3k^2+5k+3) +1}{3k+3}\right\rfloor = \left\lfloor\dfrac{9k^3 + 21k^2+19k+7}{3k+3}\right\rfloor = \left\lfloor 3k^2 + 4k + 2 + \tfrac{k +1}{3k+3}\right\rfloor = 3k^2 + 4k + 2.$$ Also, $$b_{n+2} = b_{3k+3} = \left\lfloor\dfrac{(3k+2)^2 + 3}3\right\rfloor = \left\lfloor 3k^2 + 4k+2 +\tfrac13\right\rfloor = 3k^2 + 4k + 2 = a_{3k+3}.$$
This in turn implies that $$a_{n+3} = a_{3k+4} = \left\lfloor\dfrac{(3k+3)(3k^2+7k+5) +1}{3k+4}\right\rfloor = \left\lfloor\dfrac{9k^3 + 30k^2+36k+16}{3k+4}\right\rfloor = 3k^2 + 66k + 4.$$ Also, $$b_{n+3} = b_{3k+4} = \left\lfloor\dfrac{(3k+3)^2 + 3}3\right\rfloor = 3k^2 + 6k + 4 = a_{3k+4}.$$ Notice that the fractions for $a_{n+3}$ and $b_{n+3}$ both had zero remainder on division, and therefore gave integer results without having to take the integer part. Notice also that $3k^2 + 6k + 4 = 3(k+1)^2 + 1$, which completes the inductive step.

#### hxthanh

##### New member
$a_n = 1 + \left\lfloor\dfrac{(n-1)^2}3\right\rfloor.$
...
very nice solution!

Some similar results

$a_n=1+\left\lceil\dfrac{n(n-2)}{3}\right\rceil$

$a_n=1+(2n-2)\left\lfloor\dfrac{n}{3}\right\rfloor-3\left\lfloor\dfrac{n}{3}\right\rfloor^2$

Also, by induction show that $a_n-a_{n-1}=\left\lfloor\dfrac{2(n-1)}{3}\right\rfloor$


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# If x is a positive integer, is x-1 a factor of 104?

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If x is a positive integer, is x-1 a factor of 104? [#permalink]

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22 Jan 2012, 16:44
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If x is a positive integer, is x – 1 a factor of 104?

(1) x is divisible by 3.
(2) 27 is divisible by x.

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Re: Factor of 104 [#permalink]

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22 Jan 2012, 16:59
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enigma123 wrote:
If x is a positive integer, is x – 1 a factor of 104?
(1) x is divisible by 3.
(2) 27 is divisible by x.

For me the answer is clearly B. But OA is C. Can someone please explain?

If x is a positive integer, is x – 1 a factor of 104?

(1) x is divisible by 3 --> well, this one is clearly insufficient, as x can be 3, x-1=2 and the answer would be YES but if x is 3,000 then the answer would be NO.

(2) 27 is divisible by x --> factors of 27 are: 1, 3, 9, and 27. Now, if x is 3, 9, or 27 then the answer would be YES (as 2, 8, and 26 are factors of 104) BUT if x=1 then x-1=0 and zero is not a factor of ANY integer (zero is a multiple of every integer except zero itself and factor of none of the integer). Not sufficient.

(1)+(2) As from (1) x is a multiple of 3 then taking into account (2) it can only be 3, 9, or 27. For all these values x-1 is a factor of 104. Sufficient.

Answer: C.
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Re: If x is a positive integer, is x-1 a factor of 104? [#permalink]

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22 Jan 2012, 17:08
Thanks very much buddy for shedding light on concept of ZERO.
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If x is a positive integer, is x-1 a factor of 104? [#permalink]

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22 Apr 2015, 23:12
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Hi Guys,

The question deals with the concepts of factors and multiples of a number. Its important to analyze the information given in the question first before preceding to the statements. Please find below the detailed solution:

Step-I: Understanding the Question
The question tells us that $$x$$ is a positive integer and asks us to find if $$x-1$$ is a factor of 104

Step-II: Draw Inferences from the question statement
Since $$x$$ is a +ve integer, we can write $$x>0$$. The question talks about the factors of 104. Let's list out the factors of 104.

$$104 = 13 * 2^3$$. So, factors of 104 are {1,2,4,8,13,26,52,104}, a total of 8 factors.

If $$x-1$$ is to be a factor of 104, $$2<=x<=105$$. With these constraints in mind lets move ahead to the analysis of the statements.

Step-III: Analyze Statement-I independently
St-I tells us that $$x$$ is divisible by 3. This would mean that $$x$$ can take a value of any multiple of 3. Now, all the multiples of 3 are not factors of 104. So, we can't say for sure if $$x-1$$ is a factor of 104. Hence, statement-I alone is not sufficient to answer the question.

Step-IV: Analyze Statement-II independently
St-II tells us that 27 is divisible by $$x$$ i.e. $$x$$ is a factor of 27. Let's list out the factors of 27 - {1,3,9,27}. But, we know that for $$x-1$$ to be a factor of 104, $$2<=x<=105$$. We see from the values of factors of 27, $$x$$ can either be less than 2(i.e. 1) or greater than 2 (i.e. 3,9 & 27). Hence, statement-II alone is not sufficient to answer the question.

Step-V: Analyze both statements together
St-I tells us that $$x$$ is a multiple of 3 and St-II tells us that $$x$$ can take a value of {1, 3, 9, 27}. Combining these 2 statements we can eliminate $$x=1$$ from the values which $$x$$ can take. So, $$x$$={ 3, 9, 27} and $$x-1$$ = {2, 8, 26}. We observe that all the values which $$x-1$$ can take is a factor of 104. Hence, combining st-I & II is sufficient to answer our question.

Answer: Option C

Takeaway
Analyze the information given in the question statement properly before proceeding for analysis of the statements. Had we not put constraints on the values of x, we would not have been able to eliminate x=1 from st-II analysis.

Hope it helps!

Regards
Harsh
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If x is a positive integer, is x-1 a factor of 104? [#permalink]

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20 Sep 2016, 10:28
Let's take x=30, in this case,
1. A Will be sufficient. However 30-1 is 29 is not a factor of 104.
2. 27 is also not divisible by 30. Not sufficient.

Hence, In this case is the answer E. Can anybody answer my doubt.
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If x is a positive integer, is x-1 a factor of 104? [#permalink]

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21 Sep 2016, 01:47
prashantrchawla wrote:
Let's take x=30, in this case,
1. A Will be sufficient. However 30-1 is 29 is not a factor of 104.
2. 27 is also not divisible by 30. Not sufficient.

Hence, In this case is the answer E. Can anybody answer my doubt.

I am not sure what you are trying to do here.

for statement 1 : you are considering only one value of x, which is making your case sufficient. Take x =3 and x = 6, you will get 104 divisible for x-1 = 2 but not for x-1 = 5.

Hence, it is Insufficient.

Statement 2 : We are given 27 is divisible by x. It means x is a factor of 27. The factors could be 1,3,9 and 27.

Divide 104 by each of (x-1) as 0, 2,8 and 26. You will find 104 divisible by all but 0. hence, insufficient.

On combining, we know that x cannot be 0. Hence, Answer C.
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Re: If x is a positive integer, is x-1 a factor of 104? [#permalink]

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21 Sep 2016, 03:13
prashantrchawla wrote:
Let's take x=30, in this case,
1. A Will be sufficient. However 30-1 is 29 is not a factor of 104.
2. 27 is also not divisible by 30. Not sufficient.

Hence, In this case is the answer E. Can anybody answer my doubt.

Your logic there is not clear. Why do you take x as 30? You cannot arbitrarily take x to be 30 and work with this value only. Also, how is the first statement sufficient? If x is 3, then x-1=2 and the answer would be YES but if x is 3,000 then the answer would be NO.

Please re-read the solutions above.
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Re: If x is a positive integer, is x-1 a factor of 104? [#permalink]

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04 Dec 2017, 11:54
enigma123 wrote:
If x is a positive integer, is x – 1 a factor of 104?

(1) x is divisible by 3.
(2) 27 is divisible by x.

Lets look at the prime factorisation of 104: 2^3 * 13
Thus factors of 104 = 1, 2, 4, 8, 13, 26, 52, 104
We are asked whether x-1 is one of these 8 integers, OR IS x one of these: 2, 3, 5, 9, 14, 27, 53, 105

(1) x is divisible by 3, so x could be any multiple of 3 like 9 or 27 or 54. Insufficient.

(2) 27 is divisible by x, so x is a factor of 27. Now factors of 27 are: 1, 3, 9, 27.
If x is 1, then x-1 is 0 and thus NOT a factor of 104, but if x is 3 or 9 or 27, then x-1 will take values as 2 or 8 or 26 respectively, and thus BE a factor of 104.
So Insufficient.

Combining the two statements, x has to be a multiple of 3, yet a factor of 27 also. So x could be either 3 or 9 or 27. For each of these cases, x-1 will be a factor of 104, as explained in statement 2. Sufficient. Hence C answer
Re: If x is a positive integer, is x-1 a factor of 104?   [#permalink] 04 Dec 2017, 11:54
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# At the rate of f floors per m minutes, how many floors does an elevato

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At the rate of f floors per m minutes, how many floors does an elevato  [#permalink]

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21 Nov 2015, 07:34
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At the rate of f floors per m minutes, how many floors does an elevator travel in s seconds?

(A) $$\frac{fs}{60m}$$

(B) $$\frac{ms}{60f}$$

(C) $$\frac{fm}{s}$$

(D) $$\frac{fs}{m}$$

(E) $$\frac{60s}{fm}$$

Explanation: You’re given a rate and a time, and you’re looking for distance. This is clearly a job for the rate formula. Since the rate is in terms of minutes and the time is in seconds, you’ll need to convert one or the other; it’s probably easier to convert s seconds to minutes than the rate to floors per second. Since 1 minute equals 60 seconds, s seconds equals $$\frac{s}{60}$$ minutes. Now we can plug our rate and time into the rate formula: $$r=\frac{d}{t}$$
$$\frac{f}{m}=d/\frac{s}{60}$$

Now, cross-multiply:

$$dm = \frac{fs}{60}$$

$$d=\frac{fs}{60m}$$, choice (A).
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At the rate of f floors per m minutes, how many floors does an elevato  [#permalink]

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21 Nov 2015, 14:46
gmatser1 wrote:
At the rate of f floors per m minutes, how many floors does an elevator travel in s seconds?

(A) $$\frac{fs}{60m}$$

(B) $$\frac{ms}{60f}$$

(C) $$\frac{fm}{s}$$

(D) $$\frac{fs}{m}$$

(E) $$\frac{60s}{fm}$$

Looks like a good candidate for the INPUT-OUTPUT approach.

Let's INPUT some values for f, m and s.
Let's say that f = 8 floors, m = 2 minutes, and s = 30 seconds

That is, the elevator travels at a rate of 8 floors per 2 minutes.
How many floors does an elevator travel in 30 seconds?

Well, 8 floors in 2 minutes translates to 4 floors in 1 minute, and 2 floors in 30 seconds.

So, when f = 8, m = 2, and s = 30, the answer to the question (OUTPUT) is 2 floors

Now, let's plug f = 8, m = 2, and s = 30 into each answer choice and see which one yields an OUTPUT of 2

(A) $$\frac{(8)(30)}{60(2)}$$ = 2 GREAT!

(B) $$\frac{(2)(30)}{60(8)}$$ = 1/8 ELIMINATE

(C) $$\frac{(8)(2)}{(30)}$$ = 8/15 ELIMINATE

(D) $$\frac{(8)(30)}{(2)}$$ = 120 ELIMINATE

(E) $$\frac{60(30)}{(8)(2)}$$ = some big number ELIMINATE

For more information on this question type and this approach, we have some free videos:
- Variables in the Answer Choices - http://www.gmatprepnow.com/module/gmat- ... /video/933
- Tips for the Algebraic Approach - http://www.gmatprepnow.com/module/gmat- ... /video/934
- Tips for the Input-Output Approach - http://www.gmatprepnow.com/module/gmat- ... /video/935

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Re: At the rate of f floors per m minutes, how many floors does an elevato  [#permalink]

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21 Nov 2015, 15:19
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f/m=floors per minute
f/60m=floors per one second
fs/60m=floors per s seconds
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Re: At the rate of f floors per m minutes, how many floors does an elevato  [#permalink]

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29 Sep 2017, 10:31
gmatser1 wrote:
At the rate of f floors per m minutes, how many floors does an elevator travel in s seconds?

(A) $$\frac{fs}{60m}$$

(B) $$\frac{ms}{60f}$$

(C) $$\frac{fm}{s}$$

(D) $$\frac{fs}{m}$$

(E) $$\frac{60s}{fm}$$

We have a rate of (f floors)/(m minutes) and need to determine how many floors an elevator travels in s seconds = s/60 minutes, and thus:

f/m x s/60 = fs/60m

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Re: At the rate of f floors per m minutes, how many floors does an elevato  [#permalink]

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14 Jan 2018, 23:19
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Let's let f = 60 in m = 1 minutes as it will make the calculation easy!
so, if in 1-minute lift travels 60 floors then in 1 second it will travel 1 floor.
Plugging the values as f=60,s=1,m=1 the result should be 1
So, the answer is A!

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Re: At the rate of f floors per m minutes, how many floors does an elevato   [#permalink] 13 Oct 2019, 20:17
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