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How is it not prime relatives???

Source: Indonesian MO 2022/5

October 5, 2022
number theoryINAMO 2022

Problem Statement

Let N≥2N\ge2 be a positive integer. Given a sequence of natural numbers a1,a2,a3,…,aN+1a_1,a_2,a_3,\dots,a_{N+1} such that for every integer 1≤i≤j≤N+11\le i\le j\le N+1, aiai+1ai+2…aj≢1mod  Na_ia_{i+1}a_{i+2}\dots a_j \not\equiv1\mod{N} Prove that there exist a positive integer k≤N+1k\le N+1 such that gcd⁡(ak,N)≠1\gcd(a_k, N) \neq 1
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