Problem
NT-B1-M01-P015 Greatest Divisor of Four Consecutive Terms
#15
★★★☆☆ Level 3 of 5
Find the greatest positive \(m\) such that \(m\mid n(n+1)(n+2)(n+3)\) for every integer \(n\).
Prove divisibility by \(24\), then substitute \(n=1\).
The product of four consecutive integers is always divisible by \(24\): one factor is divisible by \(3\), and the two even factors together give a factor \(8\). Thus \(24\) works. If \(m\) works for all \(n\), then at \(n=1\), it divides \(1\cdot2\cdot3\cdot4=24\). Therefore the greatest \(m\) is \(24\).
The next step after the problem on three consecutive factors.