1610.12 – Primes in Product
Let be a positive integer. Consider the product
. Name two prime numbers and one composite number that are always factors of this product, for any . Can you explain why this is true?
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.