I could use some help with this problem:
Let a, b ∈ Z (integer set) such that (a, b) = 1. Suppose that ab = $x^2$ for some x in Z (integer set). Prove that a = $y^2$ and b = $z^2$ for some y and z in Z(integer set).
Because of (a, b) = 1, I assume a and b have no common factors.
So a can be written as a = $p_1, p_2, p_3... p_s$ where p is any positive integer. Same goes for b = $q_1, q_2, q_3...q_s$ where q is any positive integer.
Since ab = $x^2$, it can be said that $x^2$ = ( $p_1, p_2, p_3... p_s$)($q_1, q_2, q_3...q_s$) = ab
And I get lost from here... I think it is only possible for this to be true if and only if all the exponent variables are even numbers. Could any help me elaborate on this?