If a, b, k, and m are positive integers, is a^k a factor of b^m?

Question: is a^k a factor of b^m --> \(a^kx=b^m\), where x is an integer? --> \(x=\frac{b^m}{a^k}\). So basically the question is: Is x an integer greater than 0?


(1) a is a factor of b --> \(ay=b\) --> \(x=\frac{a^my^m}{a^k}\) --> \(x=a^{m−k}y^m\). Now if m < k and a is not a factor of y, then x will not be an integer. Not sufficient.

Or even without any algebra: if a and b are equal to say 3 and m<k (there are less b's than a's) then a^k won't be a factor of b^m. Though if k<=m then even if a and b are not equal still a^k will be a factor of b^m as there will be enough b's for a's.


(2) k ≤ m. Not sufficient on it's own.


(1)+(2) \(x=a^{m−k}y^m\) and k<m, hence x is an integer. Sufficient. (Or again as there are more b's then a's (enough b's for a) then a^k is a factor of b^m, for example (bbb)/(aa))


Answer: C.
_________________