Discussion
Let \(p_{1}\) and \(p_{2}\) denote two arbitrary prime numbers. Which one of the following statements is correct for all values of \(p_{1}\) and \(p_{2}\)?
Answer: B
The statement that holds for all primes is that \(p_{1}p_{2}\) is not a prime number.
A prime has exactly two divisors, 1 and itself. The product \(p_{1}p_{2}\) of two primes, both at least 2, is divisible by \(p_{1}\), which is neither 1 nor the whole product. So \(p_{1}p_{2}\) is always composite, even when \(p_{1} = p_{2}\).
The others fail for some primes. \(p_{1} + p_{2}\) can be prime: 2 + 3 = 5. \(p_{1} + p_{2} + 1\) can be composite: 3 + 5 + 1 = 9. \(p_{1}p_{2} + 1\) can be composite: 3 × 5 + 1 = 16.
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