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 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:

In "what is the value" questions on Data Sufficiency, you need information that leads to one, and only one, possible answer. If the information suggests that the variable could be any multiple of a certain number, it is not sufficient.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The question asks for a specific value of an integer \(p\). To determine the value of \(p\), we need enough information to uniquely identify it. Factors of a number are integers that divide it without a remainder.
Step 2: Key Formula or Approach:
If a number has several integers as its factors, it must be a multiple of their Least Common Multiple (LCM).
Step 3: Detailed Explanation:
Analyze Statement (1): Each of the integers 2, 3, and 5 is a factor of \(p\).
This means \(p\) must be a multiple of the LCM of 2, 3, and 5.
Since 2, 3, and 5 are prime numbers, their LCM is their product: \(2 \times 3 \times 5 = 30\).
So, \(p\) is a multiple of 30. Possible values for \(p\) are 30, 60, 90, 120, and so on.
Since \(p\) can have multiple values, statement (1) is not sufficient.
Analyze Statement (2): Each of the integers 2, 5, and 7 is a factor of \(p\).
This means \(p\) must be a multiple of the LCM of 2, 5, and 7.
Since 2, 5, and 7 are prime numbers, their LCM is their product: \(2 \times 5 \times 7 = 70\).
So, \(p\) is a multiple of 70. Possible values for \(p\) are 70, 140, 210, and so on.
Since \(p\) can have multiple values, statement (2) is not sufficient.
Analyze Both Statements Together:
From statement (1), \(p\) is a multiple of 30.
From statement (2), \(p\) is a multiple of 70.
Combining them, \(p\) must be a multiple of the LCM of 30 and 70.
LCM(30, 70) = LCM(\(3 \times 10\), \(7 \times 10\)) = \(10 \times \text{LCM}(3, 7) = 10 \times 21 = 210\).
So, \(p\) is a multiple of 210. Possible values for \(p\) are 210, 420, 630, and so on.
Even with both statements, we cannot determine a unique value for \(p\).
Step 4: Final Answer:
The combined information is not sufficient to find a single value for \(p\).