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2n+1 and an+1 are relatively prime

Source: Indonesian National Science Olympiad 2018, Mathematics P1

July 6, 2018
number theoryrelatively prime

Problem Statement

Let aa be a positive integer such that gcd⁡(an+1,2n+1)=1\gcd(an+1, 2n+1) = 1 for all integer nn.
a) Prove that gcd⁡(a−2,2n+1)=1\gcd(a-2, 2n+1) = 1 for all integer nn. b) Find all possible aa.
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