Need help to solve number theory problems

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Absur Khan Siam
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Need help to solve number theory problems

Unread post by Absur Khan Siam » Mon Nov 28, 2016 10:41 am

I am trying these two problem for many days . Can anyone help me? :-
Problem no.1 : Prove that for all integer $x$ and $y$ there is no solution for $15x^2 - 7y^2 = 9$
Problem no.2 : $X$ is a two digit positive integer and $Y$ is a three digit positive integer. If $X$ is increased by $Y%$ and $Y$ is decreased by $X%$, they results same. How many solutions of $(X, Y)$ are there for this case?(from the BdMO '16 divisional)

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Thamim Zahin
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Re: Need help to solve number theory problems

Unread post by Thamim Zahin » Mon Dec 05, 2016 4:02 pm

In problem (1) take mod $5$. And remember that $y^2 \equiv 0,1,4$ $(mod $ $ 5)$.
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Thanic Nur Samin
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Re: Need help to solve number theory problems

Unread post by Thanic Nur Samin » Wed Dec 07, 2016 1:50 am

For problem 2:

Please check your statement. Missing a $ \textbf{%} $ would certainly not help people who want to answer your question.

From the question, it follows that,
\[X\left( 1+\dfrac{Y}{100}\right) = Y\left( 1-\dfrac{X}{100} \right) \]
\[XY=50Y-50X\]
\[(50-X)(50+Y)=2500\]

Note that $50-X$ is positive, as the other two are positive. Let $A=50-X$ and $B=50+Y$. $X\ge 10 \Rightarrow A\le 40$ and $100\le Y\le 999 \Rightarrow 150\le B\le 1049$.

By considering all the cases when the product of two positive integers is equal to $2500$ it is straightforward to deduce that the number of solutions is $\boxed{3}$, Namely $(X,Y)=(40,200),(45,450),(46,575)$
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jayon
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Re: Need help to solve number theory problems

Unread post by jayon » Thu Dec 08, 2016 4:22 pm

How did you get that?
\[X\left(1+\frac{y}{100}\right)=Y\left(1-\frac{X}{100}\right)\]
Last edited by Phlembac Adib Hasan on Fri Dec 09, 2016 3:22 pm, edited 1 time in total.
Reason: Latexed

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Thanic Nur Samin
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Re: Need help to solve number theory problems

Unread post by Thanic Nur Samin » Mon Dec 12, 2016 2:54 pm

Umm, $X$ is increased by $Y\%$ and $Y$ is decreased by $X\%$. The equation naturally follows.
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zarif
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Re: Need help to solve number theory problems

Unread post by zarif » Sun Jan 15, 2017 7:08 pm

In problem 2;answer is 2
(40,200),(45,450)

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