Question

If "P" is a positive integer, is \(P^4 + 7\) an odd number?
I. "P" is the smallest positive integer that is divisible by all the integers from 51 to 55, inclusive.
II. \(13^P\) is an odd number.

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II together are sufficient but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

To determine if a number is even or odd, check its prime factorization. If the prime factorization contains at least one '2', the number is even; otherwise, it is odd. The LCM of a set of integers is even if at least one of the integers in the set is even.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a "Yes/No" Data Sufficiency question about number properties (odd/even). The expression is \(P^4 + 7\). The number 7 is odd. The sum of two integers is odd if and only if one is even and one is odd. For \(P^4 + 7\) to be odd, \(P^4\) must be even. For a power of an integer (\(P^4\)) to be even, the base (\(P\)) must be even. Therefore, the question is equivalent to: "Is P an even integer?"
Step 2: Detailed Explanation:
Analyze Reconstructed Statement I: "P is the smallest positive integer that is divisible by all the integers from 51 to 55, inclusive."
This means that P is the Least Common Multiple (LCM) of the set of numbers \{51, 52, 53, 54, 55\}. To find the LCM, we need the prime factorization of each number. However, to determine if P is even or odd, we only need to check if the prime factorization of the LCM contains a '2'. The set of numbers is \{51, 52, 53, 54, 55\}. The number 52 is even (\(52 = 2 \times 26\)). The number 54 is also even (\(54 = 2 \times 27\)). Since P must be divisible by 52 (an even number), P itself must be even. Because P is even, the answer to the rephrased question "Is P an even integer?" is a definite "Yes". Therefore, Statement I is sufficient.
Analyze Reconstructed Statement II: "\(13^P\) is an odd number."
The number 13 is an odd integer. Any positive integer power of an odd number is always odd (since Odd \(\times\) Odd = Odd). This statement is true for any positive integer P, whether P is even or odd. For example, if P=1 (odd), \(13^1 = 13\) is odd. If P=2 (even), \(13^2 = 169\) is odd. Since this statement does not tell us whether P is even or odd, we cannot answer the question. Therefore, Statement II is not sufficient.
Step 3: Final Answer:
Statement I alone is sufficient, but Statement II alone is not. This corresponds to option (A).