elementary number theory - Prove that if $a^nmid b^n$ then $amid b$

Prove that if $a^n \mid b^n$ then $a\mid b$ (without use of GCD and factorization theorem).

Answer

If $a^n|b^n$, then $b^n/a^n=(\frac{b}{a})^n$ is an integer. Since $\frac{b}{a}$ is rational, this implies that $\frac{b}{a}$ is an ordinary integer, by this theorem.
Q.E.D.
notice that the use of the theorem is suggested by @CalvinLin .