Is the product of two prime numbers also a prime number?

Concept

A prime number is a natural number greater than 1 that has exactly two positive divisors: 1 and the number itself. The moment a number picks up even one more divisor, it stops being prime and becomes composite.

Application

1. Let p and q be any two prime numbers (they may be the same prime used twice, or two different primes).

2. Every divisor of p and every divisor of q is automatically a divisor of the product p × q, so 1, p, q, and p × q are all divisors of the product.

3. When p and q are different, that already gives four divisors; when p and q are the same prime, the divisors are 1, p, and p² — three divisors.

4. A prime number is allowed exactly two divisors, so a number with three or more divisors can never be prime — the product is always composite.

Cross-check

Take two different primes: 7 × 11 = 77, whose divisors are 1, 7, 11, and 77 — four divisors, so 77 is composite. Take the same prime twice: 2 × 2 = 4, whose divisors are 1, 2, and 4 — three divisors, again composite.

So the product of two prime numbers always has more than two divisors and is therefore never prime, regardless of which primes are chosen.