If A_{n}=P_{n}+1, where P_{n} is the product of the first n prime numbers, then consider the following statements:<br>1. A_{n} is <strong>ALWAYS</strong> a composite number.<br>2. A_{n}+2 is <strong>ALWAYS</strong> an odd number.<br>3. A_{n}+1 is <strong>ALWAYS</strong> an even number.<br>Which of the above statements is/are correct?
- A. 1 only
- B. 2 only
- C. 3 only
- D. 2 and 3 only ✓
Correct Answer: D. 2 and 3 only
Explanation
Since P_n contains the prime 2 for any n \geq 1, P_n is always an even number. Therefore, A_n = P_n + 1 is always odd. A_n + 2 is odd + even = odd. A_n + 1 = P_n + 2 is even + even = even. Stmt 1 is false (e.g., P_1+1 = 3, prime).
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