If $a^2|bc$, does $a|b$ or $a|c$

Question

Basically, I am trying to prove that $a^2|bc$ implies $a|b$ or $a|c$.

I can visualize it in my head (based on the fundamental theorem of arithmetic and the factors of a,b,c), however, everything I try to write it down/express it in mathematical terms fail. The closest I’ve gotten is showing $a|bc$, however, that is rather trivial.

Any help would be greatly appreciated - I am clearly missing something.

Answer 1

This will fail for the same reason that $a\mid bc$ does not directly imply that $a\mid b$ or $a\mid c$

Consider if $a=pq, b=p^2,c=q^2$ with $p,q$ both prime.