Question

What is the value of the integer \( p \)?
(1) Each of the integers 2, 3, and 5 is a factor of \( p \).
(2) Each of the integers 2, 5, and 7 is a factor of \( p \).

• 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 working with factors, consider the least common multiple (LCM) and be aware that multiple values of \( p \) can satisfy the conditions.

Answer 1

Answer: (E)

Step 1: Analyze statement (1).
Statement (1) tells us that 2, 3, and 5 are factors of \( p \).
Therefore, \( p \) is a multiple of the least common multiple (LCM) of 2, 3, and 5. The LCM of 2, 3, and 5 is 30, so \( p \) must be a multiple of 30.
However, statement (1) alone does not uniquely determine \( p \) because \( p \) could be any multiple of 30 (e.g., 30, 60, 90, etc.).
Step 2: Analyze statement (2).
Statement (2) tells us that 2, 5, and 7 are factors of \( p \).
Therefore, \( p \) is a multiple of the LCM of 2, 5, and 7. The LCM of 2, 5, and 7 is 70, so \( p \) must be a multiple of 70.
Again, statement (2) alone does not uniquely determine \( p \) because \( p \) could be any multiple of 70 (e.g., 70, 140, 210, etc.).
Step 3: Combine the two statements.
Combining both statements, we know that \( p \) is a multiple of both 30 and 70. The LCM of 30 and 70 is 210, so \( p \) must be a multiple of 210.
However, since we don’t have any additional information, we still cannot uniquely determine \( p \) because \( p \) could be any multiple of 210 (e.g., 210, 420, 630, etc.). \[ \boxed{E} \]