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Specific divisors implying perfect square

Source: Serbian MO 2017 4

April 2, 2017
modular arithmetic

Problem Statement

Let aa be a positive integer.Suppose that ∀n\forall n ,∃d\exists d, d≠1d\not =1, d≡1(modn)d\equiv 1\pmod n ,d∣n2a−1d\mid n^2a-1.Prove that aa is a perfect square.
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