Bunuel wrote:
If x is a perfect square greater than 1, what is the value of x?
(1) x has exactly 3 distinct factors.
(2) x has exactly one positive odd factor.
We are given that x is a perfect square greater than 1 and need to determine the value of x.
Statement One Alone:x has exactly 3 distinct factors.
Since we know that x is a perfect square and has 3 distinct factors, we can determine that x is the square of a prime number.
For instance, x could equal 2^2 = 4, which has distinct factors of 1, 2, and 4, or x could equal 3^2 = 9, which has distinct factors of 1, 3, and 9. Thus, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.
Statement Two Alone:x has exactly one positive odd factor.
If x has exactly one positive odd factor, then that odd factor must be 1, since 1 is a factor of every number. This leads us to the conclusion that any other factors of x must be even.
In fact, x must be the square of a power of 2. For example, x could be (2^1)^2 = 2^2 = 4 or (2^2)^2 = 4^2 = 16. In the former case, the distinct factors of x are 1, 2 and 4; exactly one of them is odd. In the latter case, the distinct factors of x are 1, 2, 4, 8, and 16; again, exactly one of them is odd. Because x can equal, for example, 4 or 16, we don’t have a unique value for x. We can eliminate answer choice B.
Statement One and Two Together:From our analysis of statement one, we determined that x is the square of a prime number, and from our analysis of statement two, we determined that x is the square of a power of 2. Since 2 is the only prime that is a power of 2, x must equal 2^2 = 4.
Answer: C