Generalization of Euclid's Lemma for Powers of Coprimes

Question

The generalized Euclid's lemma states that for $a,b,c\in \mathbb{Z}$, if $a|bc$ and $\gcd(a,b)=1$, then $a|c$. Now, from this, can we prove that for $i,j\in \mathbb{N^*}$ if $\gcd(a,b)=1$ and $a^i|b^jc$, then $a^i|c$? I actually even want to know if it's true if we let $i,j \in \mathbb{Q}$ provided $a^i,b^j \in \mathbb{Z}$.

Answer 1

For your first statement:

Let $m = a^i$ and $n = b^j$.

Then by the unique prime decomposition, and the fact that $a,b$ have no primes in common, we conclude that $m,n$ have no primes in common either. Thus: $gcd(m,n) = (gcd(a,b))^{\min({i,j})} = gcd(a,b) = 1$. But then:

$$m \vert nc \Rightarrow m \vert c$$

For the case where $i, j \in \mathbb{Q}$ we work similarly, as the fact that $a,b$ have no primes in common still implies that $m,n$ don't have any primes in common either. Granted, some of their powers in their corresponding decompositions may be reduced, but no common primes will show up, even after "fractional" (rational) exponentiation.