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.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When solving for prime factors, combine the information about divisibility to identify all the factors of the number.

Answer 1

The Correct Option is C

Step 1: Analyze statement (1).
Statement (1) tells us that \( \frac{k}{15} \) is an integer, meaning that \( k \) is divisible by 15. Since 15 can be factored as \( 3 \times 5 \), we know that \( k \) is divisible by 3 and 5. However, this alone does not tell us if \( k \) has at least three different prime factors. For example, if \( k = 15 \), then it has exactly two prime factors, 3 and 5.
Step 2: Analyze statement (2).
Statement (2) tells us that \( \frac{k}{10} \) is an integer, meaning that \( k \) is divisible by 10. Since 10 can be factored as \( 2 \times 5 \), we know that \( k \) is divisible by 2 and 5. However, this alone does not tell us if \( k \) has at least three different prime factors.
Step 3: Combine statements (1) and (2).
From statement (1), \( k \) is divisible by 15, and from statement (2), \( k \) is divisible by 10. Combining these, \( k \) is divisible by \( 2 \times 3 \times 5 = 30 \), so \( k \) has at least three different prime factors: 2, 3, and 5.
\[ \boxed{C} \]