Let 𝑝1 and 𝑝2 denote two arbitrary prime numbers. Which one of the following…

Answer: The product of the two primes is not a prime number.

Reason: If p1 and p2 are prime numbers (both at least 2), then p1·p2 has divisors p1 and p2 and is greater than 1, so p1·p2 is composite and therefore not prime.

Counterexamples showing the other statements fail in general:

• The claim that the sum of the primes is not prime is false: for example 2 + 3 = 5, which is prime.

• The claim that the sum of the primes plus one is always prime is false: for example 2 + 3 + 1 = 6, which is not prime.

• The claim that the product of the primes plus one is always prime is false: for example 3·5 + 1 = 16, which is not prime (though sometimes p1·p2 + 1 can be prime, e.g. 2·3 + 1 = 7).