Problem
NT-B1-M12-P006 Set 2. Four Consecutive Integers
#6
★★☆☆☆ Level 2 of 5
Prove that \(24\mid n(n+1)(n+2)(n+3)\) for every integer \(n\).
Prove divisibility by \(8\) and by \(3\).
Among four consecutive integers, one is divisible by \(3\). Also there are two even numbers, one of which is divisible by \(4\), so the product is divisible by \(8\). Since \(8\) and \(3\) are coprime, the product is divisible by \(24\).
Stronger than the usual \(6\mid n(n+1)(n+2)\).