**GCD**and

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

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?

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:)....

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:)....

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.

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**

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

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.

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**