If $\gcd(a, b) = p$, where $p$ is a prime. Then what are possible values of $\gcd(a^3, b^2)$?
I have already calculated that $\gcd(a^2, b^2) = p^2$. $\gcd(a^2, b) = p$ or $p^2$, $\gcd(a^3, b) = p$ or $p^2$ or $p^3$. Can anybody give me any idea? how will I find the possible value of $\gcd(a^3, b^2)$?
Since $p$ is prime, each of $a$ and $b$ contain at least one factor of $p$, and one of them contains only one.
Case 1: If $b$ contains only one factor of $p$ regardless of how many factors of $p$ are contained in $a$, then $a^3$ contains at least $3$ factors, and $b^2$ contains only two factors of $p$, so $\gcd (a^3,b^2)=p^2$.
Case 2: If $a$ contains only one factor of $p$ and $b$ contains more than one factor of $p$, then $a^3$ contains $3$ factors of $p$, and $b^2$ contains at least $4$ factors of $p$, so $\gcd (a^3,b^2)=p^3$.