udaymathapati wrote:

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

(1) a is a factor of b.

(2) k = m

We are given that a, b, k, and m are positive integers, and we need to determine whether a^k is a factor of b^m.

Statement One Alone:

a is a factor of b.

Although we know that a is a factor of b, we still cannot determine whether a^k is a factor of b^m.

For example, if a = 2, b = 4 (2 is factor of 4), k = 1, and m = 1, then 2^1 = 2 is a factor of 4^1 = 4.

However, if a = 2, b = 4, k = 3, and m = 1, then 2^3 = 8 is not factor of 4^1 = 4. Statement one alone is not sufficient.

Statement Two Alone:

k = m

Since we know neither the values of a and b nor the relationship between a and b, statement two alone is not sufficient.

Statements One and Two Together:

From statement one, we know a is a factor of b, and from statement two, we know k = m.

In order for a^k to be a factor of b^m, (b^m)/(a^k) = integer.

Since k = m, we can write (b^m)/(a^k) as (b^m)/(a^m). Now let’s simplify:

(b^m)/(a^m) = (b/a)^m

Since a is a factor of b, b/a is an integer; thus:

(b/a)^m = integer^m = integer

Thus, a^k is indeed a factor of b^m and the two statements together are sufficient.

Answer: C

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