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Sum of remainders

Source: Iberoamerican Olympiad 2005

September 29, 2005
number theory solvednumber theory

Problem Statement

Denote by a mod ba \bmod b the remainder of the euclidean division of aa by bb. Determine all pairs of positive integers (a,p)(a,p) such that pp is prime and a mod p+a mod 2p+a mod 3p+a mod 4p=a+p. a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p.
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