Exercise 1092Combinatorics › Pigeonhole principleL1Added 3 Nov 2026

Consecutive integers whose sum is divisible

#pigeonhole#consecutive sum#divisible

Statement

Show that among any $n$ consecutive integers, one is divisible by $n$ .

Hint

Residues are $0,1,\ldots,n-1$ — exactly one of each.

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Residues. In any $n$ consecutive integers, the residues mod $n$ are $\{0,1,\ldots,n-1\}$ in some order (all distinct, exactly $n$ values).

Conclusion. Exactly one has residue $0$ , so is divisible by $n$ .