Problem
ALG-B1-M09-P006 Fifth power
#6
★★☆☆☆ Level 2 of 5
Prove that \(n^5-n\) is divisible by \(5\) for every integer \(n\).
Consider residues of \(n\) modulo \(5\).
It is enough to check \(n\equiv0,1,2,3,4\pmod5\). We get \(n^5\equiv n\pmod5\) for each residue: \(0^5=0\), \(1^5=1\), \(2^5=32\equiv2\), \(3^5=243\equiv3\), \(4^5\equiv(-1)^5\equiv-1\equiv4\). Hence \(n^5-n\equiv0\pmod5\).
The first deliberate check of all residues.