Newaz111 wrote: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 ≤ m

Statement 1: a is a factor of b
Test one case that also satisfies statement 2.
Case 1: a=2, b=2, k=1, m=1
In this case, a^k = 2 and b^m = 2.
Since a^k=2 is a factor of b^m=2, the answer to the question stem is YES.

Test one case that DOESN'T also satisfy statement 2.
Case 2: a=2, b=2, k=2, m=1
In this case, a^k = 4 and b^m = 2.
Since a^k=4 is NOT a factor of b^m=2, the answer to the question stem is NO.

Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.

Statement 2: k ≤ m
Case 1 also satisfies statement 2.
In Case 1, the answer to the question stem is YES.

Case 3: a=2, b=1, k=1, m=1
In this case, a^k = 2 and b^m = 1.
Since a^k=2 is NOT a factor of b^m=1, the answer to the question stem is NO.

Since the answer is YES in Case 1 but NO in Case 3, INSUFFICIENT.

Statements combined:
Statement 1 implies that b is a multiple of a:
Let b = ax, where x is a positive integer.
Statement 2 implies that m ≥ k.
Let m = k + y, where y is a nonnegative integer.
Thus:
b^m = (ax)^(k+y) = (ax)^k * (ax)^y = (a^k)(x^k)(a^y)(x^y).
The value in red indicates that a^k is a factor of b^m.
SUFFICIENT.

The correct answer is C.

Now i got it...
Thank you.

Now i got it...
Thank you.

Now i got it...
Thank you.

Now i got it...
Thank you.

Now i got it...
Thank you.

Now i got it...
Thank you.

Great explanation Mr GMATGuruNY.

fond of your way to approach the answer.

Thank you Rich..You are right, i have to practice more..