k is a positive integer. Is k prime? (1) At least one number in the set {1, k, k + 7} is prime. (2) k is odd. A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) Statement (2) by itself is obviously not sufficient. An odd number can be prime, e.g. 3, 5, 7. An odd number can be composite (not prime), e.g. 9, 15, 35.

Let’s consider statement (1) by itself. At least one number in the set {1, k, k + 7} is prime. 1 is NOT prime. So either k or k + 7 is prime (or both). Let’s try some small values. Obviously, k can be prime, e.g. 3. Or k + 7 can be prime, e.g. k + 7 = 11. In this case k itself is NOT prime, k = 11 – 7 = 4. So k can be both: prime and composite. Therefore statement (1) by itself is NOT sufficient.

If we use the both statements together, then statement (1) implies that either k or k + 7 is prime (or both). Statement (2) implies that k is odd, so k + 7 must be even. The only even prime number is 2. But k + 7 is greater than 2, so it is NOT prime. Therefore k must be prime. Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient. The correct answer is C. ---------- In the case of k = 1, which satisfies both the statements we do not know if k is prime.

Statement 1 doesn't say that the set contains unique/distinct values.

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