Algebra Proof including relative primes.

Question

Let $c \in Z$ and $ \in a,b \in Z^{+}$ Let $c^2=ab$ suppose $(a,b)=1$

Show that there exist integers x,y s.t. $a=x^2$ and $b=y^2$

Any hints / directions will be helpful.

I know that I can write $(a,b)=1 \implies ax+by=1$

Answer 1

Multiplying $b$ in both members of $1=(a,b)$ we get, $$ b=(ab,b^2)=(c^2,b^2)=(c,b)^2. $$ The same way we can get $$ a=(a^2,ba)=(a^2,c^2)=(a,c)^2 $$