Question

If m and n are positive integers such that the greatest common factor of
`m^(2) n^(2)` and
`m n^(3)` is 45, then which of the following could n equal?

(A) 3
(B) 5
(C) 9
(D) 15
(E) 45

Answer 1

the greatest common factor of
`m^(2) n^(2)` and
`m n^(3)`

is

gcd(m*m*n*n, m*n*n*n) =m*n*n=
`m n^(2)`

what we did, was to see how many m's and how many n's, each expression have, and then pick the smallest of each letter (factor).

the reason is that the common factor, must have enough m's and n's to divide the m's and n's of both expressions

so

`45=m n^(2)`
also

`45=5*9=5*3^(2)`

thus, n is equal to 3


Answer: A)3