BdMO National 2013: Junior 9

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
BdMO
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BdMO National 2013: Junior 9

Unread post by BdMO » Fri Jan 10, 2014 1:29 am

The ratio of GCD and LCM of two integers is $1: 36$ and sum of the integers is $5460$. What is the difference between these two integers?

Tahmid
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Re: BdMO National 2013: Junior 9

Unread post by Tahmid » Sun Jan 12, 2014 12:38 am

let two numbers are a and b.
and let $\left ( a,b \right )=x \Leftrightarrow a=xa_{1}$ and $b=xb_{1}$ where $\left ( a_{1},b_{1} \right )=1$
so,$\left [ a,b \right ]=xa_{1}b_{1}$

but given that, \[ \frac{(a,b)}{[a,b]}=\frac{1}{36} \]
\[\frac{x}{xa_{1}b_{1}}=\frac{1}{36}\Leftrightarrow \frac{1}{a_{1}b_{1}}=\frac{1}{36}\Leftrightarrow a_{1}b_{1}=36\]

so now we get,$a_{1}b_{1}=36$ and $(a_{1},b_{1})=1$
the pair which satisfy these two property of $a_{1}$ and $b_{1}$ is $(4,9)$; that means $a_{1}=4$ and $b_{1}=9$

$\therefore a=4x$ and $b=9x$
given that $a+b=5460$
so, $a+b=5460\Leftrightarrow 4x+9x=5460\Leftrightarrow 13x=5460\Leftrightarrow x=420$

finally $x=420;a_{1}=4;b_{1}=9$

so $a=420*4=1680$ and $b=420*9=3780$
and difference $=3780-1680=2100$......(ans:).... :)

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Labib
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Re: BdMO National 2013: Junior 9

Unread post by Labib » Sun Jan 12, 2014 2:07 am

Another possible pair satisfying the conditions for $(a_1, b_1)$ is $(1,36)$.
Although it does not lead to any significant discovery, ignoring it might mean you losing out on some easy points for the problem.
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Learn how to write equations, and don't forget to read Forum Guide and Rules.


"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes

Tahmid
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Joined: Wed Mar 20, 2013 10:50 pm

Re: BdMO National 2013: Junior 9

Unread post by Tahmid » Sun Jan 12, 2014 9:46 am

i noticed that. but for pair (1,36), our next equation comes 37x=5460. but 37 does not divide 5460. then x becomes fraction. but x must be a integer. so (1,36) is not possible. i forget to include these steps in my solution :?

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Labib
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Re: BdMO National 2013: Junior 9

Unread post by Labib » Mon Jan 13, 2014 7:31 pm

Yes, that pair does not lead us to a solution. But it is a good practice to include these small details in your solution.
Otherwise you will lose 1 or 2 points in BdMO/BdMC/IMO. It was just a heads up. :)
I request, if it's still possible, please hide the solutions. It helps others.
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.


"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes

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