Question

Find two atural numbers whose difference us 66 and lcm is 360

Answer 1

Hey.

Here is the answer.

Let the GCD be x

then 2 numbers are mx,nx

without loss of generality let m > n

now LCM = mnx ( as m and n are copimes)

so difference = (m-n) x = 66

mnx = 360

as product mnx is 360 so 11 does not divide product of m , n and x and hence it does not divide x because if 11 divides x then it divides mnx or 360

now 11 does not devide x so 11 devides (m-n)

so, m-n is multiple of 11

Lets say m-n = 11 => x = 6

so mn = 60

m-n = 11
mn = 60

can be solved to give m = 15, n= 4 so numbers are 90 and 24


if m-n = 22 then x= 3 and

so mn = 120 does not have integer solution

m-n = 33 => x = 2 so mn = 180 does not have integer solution

m-n = 66 => x = 1 so mn = 360 does not have integer solution

So, solution is (90,24).

Thanks.