Problem
NT-B2-M12-P009 Checking by Residues
#9
★★★☆☆ Level 3 of 5
Let \(f\in\mathbb Z[x]\), \(m\ge1\). Prove: if \(m\mid f(r)\) for all \(r=0,1,\ldots,m-1\), then \(m\mid f(n)\) for all integers \(n\). Apply this to prove \(6\mid n^3-n\).
Every \(n\) is congruent to one residue \(r\).
Let \(n\equiv r\pmod m\), \(0\le r
One may discuss when residue checking is practical and when factorisation is better.