**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**

- Swapnil Barua
**Posts:**14**Joined:**Tue Apr 13, 2021 2:55 pm

In 2 numbers GCD their common factor remains multiple of those numbers and in LCM it remains with the common factors and other factors remains multiple. So in the ratio of GCD and LCM common factors cancel out and remains the others factors without those common factors.

Now, 36=2×2×3×3 these 4 factors can be of similar numbers [case 1] or 2×2 factor is of 1 number and 3×3 is of another.[case 2]

First take these 4 factors are of similar numbers, If 1 number is 36x another is x

According to the question

36x+x= 5460

or, 37x= 5460

or,x= 147.57 [ Not an integer]

Now take the 2nd case,

If 1 number is 4x another is 6x

According to the question

4x+6x=5460

or, 10x= 5460

or, x= 546

Difference=6x-4x=2x= 2× 546= 1092

Now, 36=2×2×3×3 these 4 factors can be of similar numbers [case 1] or 2×2 factor is of 1 number and 3×3 is of another.[case 2]

First take these 4 factors are of similar numbers, If 1 number is 36x another is x

According to the question

36x+x= 5460

or, 37x= 5460

or,x= 147.57 [ Not an integer]

Now take the 2nd case,

If 1 number is 4x another is 6x

According to the question

4x+6x=5460

or, 10x= 5460

or, x= 546

Difference=6x-4x=2x= 2× 546= 1092