Problem
ALG-B1-M05-P020 Coefficient Before a Product
#20
★★★★★ Level 5 of 5
Find all real \(t\) such that \(t\,n(n+3)(n+6)\) is an integer for every positive integer \(n\).
Inspired by regional olympiad method · 2011 · Grade 9 · Problem 5
Substitute several small \(n\) to restrict \(t\).
For \(n=1\), \(28t\in\mathbb Z\); for \(n=2\), \(80t\in\mathbb Z\); for \(n=3\), \(162t\in\mathbb Z\). The greatest common divisor of \(28,80,162\) is \(2\), so \(2t\in\mathbb Z\).
Conversely, \(n(n+3)(n+6)\) is always even: if \(n\) is even this is clear, and if \(n\) is odd then \(n+3\) is even. Thus every \(t=\frac{k}{2}\), \(k\in\mathbb Z\), works. Answer: \(t=\frac{k}{2}\).
Architecture preserved: several substitutions give the necessary denominator, then sufficiency is proved.