I'm working on a problem and I would like to use this fact, but I don't have anything to write on so I am having trouble figuring it out.
Is it always true that $\gcd(a, b) = \gcd(a^n, b^n)$?
I know it has to be true for prime numbers. I'm not sure about composite ones.
Only if they're coprime to begin with, that is, $\gcd(a, b) = 1$. If the the two numbers share a common prime factor, $p$, then $a^2$ is divisible by $p^2$ and so is $b^2$.
An example to make it clearer: $\gcd(4, 14) = 2$ and $\gcd(16, 196) = 4$.