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Find the minimum positive integer

Source: 2013 China Mathematical Olympaid P6

January 14, 2013

abstract algebramodular arithmeticnumber theory proposednumber theory

Problem Statement

Let

m,n

be positive integers. Find the minimum positive integer

N

which satisfies the following condition. If there exists a set

S

of integers that contains a complete residue system module

m

such that

| S | = N

, then there exists a nonempty set

A \subseteq S

so that

n\mid {\sum\limits_{x \in A} x }