Given three positive integers $a,b,c$ and I want to find the smallest positive integers $a', b', c'$ such that $$ \frac{a^2}{b} = \frac{a'^2}{b'} \quad \text{and} \quad \frac{a^3}{c} = \frac{a'^3}{c'} $$ in other words I want to find the largest integer $d$ such that $$ d\vert a, \quad d^2 \vert b \quad \text{and} \quad d^3 \vert c $$
I wonder wether I can find $d$ using a fast algorithm reliying only divisibility, root extraction or computing gcds (greatest common divisor), without relying in factorization or iteration over the possible divisors.
More generally I would like to know a polynomial time algorithm to reduce in the same way the quotients
$$ \frac{a^2}{b}, \frac{a^3}{c}, \frac{a^4}{d}, \dots , \frac{a^t}{m}$$
where $a,b,c,\dots,m$ and $t$ are positive integers.
How about looking for the gcd $g$ of $a^6$, $b^3$ and $c^2$ and then taking the "truncated 6-th root" (or whatever you would call it) of that, i.e. take $d$ to be the largest integer such that $d^6\,|\,g\;$?