Problem Solving

I have this question:

if a, b, k, and m are positive integers, is (a)^k a factor of (b)^m?

1. a is a factor of b
2. k is less than or equal to m

I think statement (1) is sufficient. But the answer is C (both statements together are sufficient, but neither statement alone is sufficient).

Thanks for your explanation.

The question is asking whether or not we can have

b^m / a^k without a remainder

b / a without a remainder is not enough

what if m <k> 81/243

so we need to know both A and B => answer is C

Is a more concrete steps possible here???

Is a^k is a factor of b^m is the same as asking, Is b^m / a^k = integer
Given a, b, k, and m are positive integers

Stmt I

a*k = b

So we can rewrite the question as: Is (ak) ^ m / a^k an integer

a^m * k^m / a^k

k^m is always going to be an integer but we dont know if a^m/a^k is going to be an integer since we have no idea about m and k and if not there is no way to tell if k^m is divisible by a^m/a^k

INSUFF

Stmt II

k<m

Easily pick numbers and disprove this choice
b=4 m=1 a=3 k=2 NO
b=4 m=2 a=2 k=1 YES

INSUFF

Together:

Lets use what we did in Stmt I

Is a^m * k^m / a^k or Is a^m/a^k an integer since k^m is always going to an integer and we know integer*integer = integer

Since m>k, a^m/a^k is an integer therefore

a^m/a^k * k^m

= integer*integer
= integer

Therefore a^k is a factor of b^m

Hope this helps!

C