Exercise 1090Combinatorics › Pigeonhole principleOlympiadAdded 3 Nov 2026

Sum of digits

#pigeonhole#digits#sums

Statement

Show that among any $82$ positive integers, some two have the same digit sum mod $10$ .

Hint

Digit sum mod $10$ has only $10$ possible values. $82 = 10\cdot 8+2$ .

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Classes. Digit sum mod $10$ has $10$ possibilities.

Pigeonhole. $82$ integers, $10$ classes: some class has $\ge\lceil 82/10\rceil=9$ integers.

Conclusion. Two integers have the same digit sum mod $10$ (in fact, at least $9$ do).