if $\gcd(a,b)=1$ and $c^2$ divides $ab$, prove that $c^2$ divides $a$ or $b$.

Question

I have searched for help, but all explanations are not detailed enough. I know that if $c^2$ divides $ab$ then there exists an $n$ such that $ab=nc^2$. I also state that since $\gcd(a,b)=1$ then there is a unique factorization such that $(a_1\cdots a_i)(b_1\cdots b_j) = n(c_1\cdots c_k)^2$. This is where I get stuck.

Answer 1

This statement is not true.

If $a$ and $b$ are squares greater than $1$ such that $\gcd(a,b)=1$ (take for instance $a=2^2$ and $b=3^2$).

Then take $c=\sqrt{ab}$, you have obviously $$c\mid ab$$ but $c\nmid a$ and $c\nmid b$.