Question

If “P” is a positive integer, is \( P^4 + 7 \) an odd number?
(1) “P” is the smallest integer such that it is divisible by all the integers from 51 to 55, inclusive.
(2) \( 13^P \) is an odd number.

• 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 but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

When analyzing divisibility problems, check the parity (odd or even) of the given terms and their relations.

Answer 1

The Correct Option is C

Step 1: Analyzing Statement 1.
Statement 1 tells us that “P” is the smallest integer divisible by the integers from 51 to 55. We can calculate the least common multiple (LCM) of these numbers, but this information does not help us determine the parity (odd/even) of \( P^4 + 7 \), making statement 1 alone insufficient.
Step 2: Analyzing Statement 2.
Statement 2 tells us that \( 13^P \) is odd. Since 13 is odd, any power of 13 will also be odd, regardless of whether \( P \) is odd or even. This alone doesn't provide enough information to determine if \( P^4 + 7 \) is odd or even. Thus, statement 2 alone is insufficient.
Step 3: Combining Both Statements.
By combining both statements, we know that “P” must be divisible by 51, 52, 53, 54, and 55, which forces “P” to be an even integer. Since \( P^4 \) is even, \( P^4 + 7 \) is odd. Thus, both statements together are sufficient to answer the question.
Step 4: Conclusion.
The correct answer is (C).