Bunuel 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.
Given: a, b, k, and m are positive integers Target question: Is a^k a factor of b^m? Statement 1: a is a factor of b. This information seems partially useful. However, since we know nothing about the exponents, a^k and b^m can be practically any magnitude.
To see what I mean consider these two possible cases that satisfy statement one:
Case a: a = 2, b = 4, k = 1 and m = 2. In this case, a^k = 2^1 = 2, and b^m = 4^2 = 16. In this case, the answer to the target question is
YES, a^k IS a factor of b^mCase b:a = 2, b = 4, k = 5 and m = 1. In this case, a^k = 2^5 = 32, and b^m = 4^1 = 4. In this case, the answer to the target question is
NO, a^k is NOT a factor of b^mSince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: k ≤ mSince we have no information about a and b, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 1: if a is a factor of b, then we can also say that
b is a multiple of a.
So, let's say
b = ax (for some positive integer)
Statement 2: k ≤ m
So, let's say m =
k + n, where n is some integer greater than or equal to 0 (this satisfies the condition that k ≤ m).
This means, we can write: b^m = (
ax)^(
k + n) = [(ax)^k][(ax)^n]
= [(a^k)(x^k)][(a^n)(x^n)]
= (
a^k)[(x^k)(a^n)(x^n)]
Since we can express b^m as a multiple of
a^k, we can be certain that
a^k IS a factor of b^m
The answer to the target question is
YES, a^k IS a factor of b^mSince we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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