What are the possible values for $\gcd(a^2, b)$ if $\gcd(a, b) = 3$? I was looking back at my notes on number theory and I came across this question.
Let $a$, $b$ be positive integers such that $\gcd(a, b) = 3$. What are the possible values for $\gcd(a^2, b)$?
I know it has to do with their prime factorization decomposition, but where do I go from here?

If $p$ is a prime, and $p|a^2$, then $p|a$; thus, if $p|a^2$ and $p|b$, then $p|a$ and $p|b$, hence $p|\gcd(a,b) = 3$. So $\gcd(a^2,b)$ must be a power of $3$.
Also, $3|a^2$ and $3|b$, so $3|\gcd(a^2,b)$; so $\gcd(a^2,b)$ is a multiple of $3$.
If $3^{2k}|a^2$, then $3^k|a$ (you can use prime factorization here); so if $3^{2k}|\gcd(a^2,b)$, then $3^k|\gcd(a,b) = 3$. Thus, $k\leq 1$. That is, no power of $3$ greater than $3^2$ can divide $\gcd(a^2,b)$.
In summary: $\gcd(a^2,b)$ must be a power of $3$, must be a multiple of $3$, and cannot be divisible by $3^3=27$. What's left? Now give examples to show all of those possibilities can occur.
