Problem
NT-B1-M01-P006 Three Consecutive Integers
#6
★★☆☆☆ Level 2 of 5
Prove that \(6\mid n(n+1)(n+2)\) for every integer \(n\).
Find a factor divisible by \(2\) and a factor divisible by \(3\).
Among three consecutive integers, one is divisible by \(3\), and at least one is even. Thus the product is divisible by \(3\) and by \(2\). Since \(\gcd(2,3)=1\), the product is divisible by \(6\).
Basic olympiad technique for consecutive integers.