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Part of 2015 Benelux

Problems(1)

BxMo 2015, Problem 1

Source: Benelux Mathematical Olympiad 2015, Problem 1

Determine the smallest positive integer

q

with the following property: for every integer

m

1\leqslant m\leqslant 1006

, there exists an integer

n

\dfrac{m}{1007}q<n<\dfrac{m+1}{1008}q