TL;DW: proof by induction. Basis is trivial: 0⁵ = 0. Inductive step: Assume that n⁵ - n is divisible by 10. We'll show that (n + 1)⁵ - (n + 1) is divisible

TL;DW: proof by induction. Basis is trivial: 0⁵ = 0. Inductive step: Assume that n⁵ - n is divisible by 10. We'll show that (n + 1)⁵ - (n + 1) is divisible by 10, too. (n + 1)⁵ - (n + 1) = n⁵ + 5n⁴ + 10n³ + 10n² + 5n + 1 - n - 1 = (n⁵ - n) + 10(n³ + n²) + 5n(n³ + 1) Notice that: * (n⁵ - n) is divisible by 10, as per the inductive hypothesis; * 10(n³ + n²) is obviously divisible by 10; * either n or n³+1 is even, so 5n(n³ + 1) is divisible by 10. QED.