BdMO National 2013: Junior 9

BdMO
Posts: 134
Joined: Tue Jan 18, 2011 1:31 pm

BdMO National 2013: Junior 9

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
Posts: 110
Joined: Wed Mar 20, 2013 10:50 pm

Re: BdMO National 2013: Junior 9

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:)....

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm

Re: BdMO National 2013: Junior 9

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.
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

Tahmid
Posts: 110
Joined: Wed Mar 20, 2013 10:50 pm

Re: BdMO National 2013: Junior 9

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

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm
Please Install $L^AT_EX$ fonts in your PC for better looking equations,