1610.12 – Primes in Product

Let $x$ be a positive integer. Consider the product

$x(x + 1)(x + 2)$ . Name two prime numbers and one composite number that are always factors of this product, for any $x$ . Can you explain why this is true?

Solution

The prime factors are 2 and 3; the composite factor is 6.

In any three consecutive whole numbers, one number will be a multiple of 3 and at least one will be a multiple of 2. So 2, 3, and 6 will always divide the product of three consecutive positive integers.