Question

Does the integer k have at least three different positive prime factors?
(1) \( \frac{k}{15} \) is an integer.
(2) \( \frac{k}{10} \) is an integer.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

When a number is a multiple of two other numbers, it must be a multiple of their LCM. Finding the prime factors of the LCM is often the key to solving problems like this one.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question is about the properties of integers, specifically their prime factors. A prime factor is a prime number that divides an integer exactly. The question is a Yes/No question: does the prime factorization of \(k\) contain at least three distinct prime numbers?
Step 2: Detailed Explanation:
Analyze Statement (1): \( \frac{k}{15} \) is an integer.
This means \(k\) is a multiple of 15. The prime factorization of 15 is \(3 \times 5\). So, the prime factorization of \(k\) must include at least 3 and 5. This guarantees two different prime factors. But does it guarantee a third?

Case 1 (Answer "No"): If \(k = 15\), its prime factors are 3 and 5. This is only two different prime factors. The answer is "No".

Case 2 (Answer "Yes"): If \(k = 30\) (which is \(15 \times 2\)), its prime factors are 2, 3, and 5. This is three different prime factors. The answer is "Yes".

Since we can get both "Yes" and "No" answers, statement (1) is not sufficient.
Analyze Statement (2): \( \frac{k}{10} \) is an integer.
This means \(k\) is a multiple of 10. The prime factorization of 10 is \(2 \times 5\). So, the prime factorization of \(k\) must include at least 2 and 5. This guarantees two different prime factors. But does it guarantee a third?

Case 1 (Answer "No"): If \(k = 10\), its prime factors are 2 and 5. This is only two different prime factors. The answer is "No".

Case 2 (Answer "Yes"): If \(k = 30\) (which is \(10 \times 3\)), its prime factors are 2, 3, and 5. This is three different prime factors. The answer is "Yes".

Since we can get both "Yes" and "No" answers, statement (2) is not sufficient.
Analyze Both Statements Together:
From statement (1), \(k\) is a multiple of 15. From statement (2), \(k\) is a multiple of 10. If \(k\) is a multiple of both 10 and 15, it must be a multiple of their least common multiple (LCM). LCM(10, 15) = LCM(\(2 \times 5\), \(3 \times 5\)) = \(2 \times 3 \times 5 = 30\). So, \(k\) must be a multiple of 30. The prime factorization of 30 is \(2 \times 3 \times 5\). Any multiple of 30 will have at least 2, 3, and 5 as its prime factors. For example, if \(k = 60 = 2^2 \times 3 \times 5\), the distinct prime factors are still 2, 3, and 5. Therefore, \(k\) is guaranteed to have at least three different positive prime factors. The answer is always "Yes". The statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but combined they are sufficient.