Problem 5 (10 points): Let $a,b$ be integers such that $g.c.d.(a,b) = p$ where $p$ is prime. Find $g.c.d.(a^2,b^2)$.
I've found that $g.c.d. (a^2,b^2) = p^2$ when using examples for $(a,b)$ like $(9,12)$, $(34,85)$, and $(14,21)$ whose gcd's are primes. I could put my answer down as $g.c.d. (a^2,b^2) = p^2$ and probably get the answer right but I would really like to find that through proofs rather than examples. Any help is appreciated.