Problem
NT-B1-M01-P014 Greatest Divisor of Three Consecutive Terms
#14
★★★☆☆ Level 3 of 5
Find the greatest positive \(m\) such that \(m\mid n(n+1)(n+2)\) for every integer \(n\).
First prove that \(6\) always works, then substitute \(n=1\).
For every \(n\), the product of three consecutive integers is divisible by \(6\), so \(m=6\) works. If some \(m\) works for all \(n\), then at \(n=1\), it divides \(1\cdot2\cdot3=6\). Hence the greatest possible \(m\) is \(6\).
Students must see both parts: existence and maximality.