Ok, considering the quotient
Q = \frac{b^n}{a^n} = \frac{q_1^{nj_1} \cdots q_{\ell}^{nj_{\ell}}}{p_1^{ni_1}\cdots p_k^{ni_k}}
we have, since Q \in\mathbb{Z}, that every prime factor in the denominator must also appear at least that many times in the numerator (I'm a little unsure about...
Thanks for the reply!
I was wondering if the prime factorization would be of any use, since it seems like we need to prove something about the quotient (that its nth root is an integer).
I can write a = p_1^{i_1} \cdots p_k^{i_k} and b = q_1^{j_1} \cdots q_{\ell}^{j_{\ell}}, where all the p's...
Hey, I'm a little stumped on this basic number theory question. The solution is probably staring me in the face, but for some reason it's eluding me...
If a^n | b^n, prove that a|b.
So, say that a^n \cdot x = b^n for some integer x, there's not a lot I can go to from there. We do get that a |...