Problem
NT-B1-M12-P021 Set 5. Sum of Powers
#21
★★★★☆ Level 4 of 5
Let \(n\) be an odd positive integer. Prove that \(n\mid1^n+2^n+\cdots+(n-1)^n\).
Group \(a\) with \(n-a\).
For each \(a=1,\ldots,n-1\), the number \(n-a\) also appears. Since \(n\) is odd, \(a^n+(n-a)^n\equiv a^n+(-a)^n=0\pmod n\). All terms split into such pairs, so the whole sum is divisible by \(n\).
Nice problem on symmetry of residues.