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a^{p_i}= 1 mod p_{i+1} for different odd primes given

Source: 2011 Belarus TST 3.1

November 7, 2020
number theoryoddprimesdivisible

Problem Statement

Given natural number a>1a>1 and different odd prime numbers p1,p2,...,pnp_1,p_2,...,p_n with ap1≡1a^{p_1}\equiv 1 (mod p2p_2), ap2≡1a^{p_2}\equiv 1 (mod p3p_3), ..., apn≡1a^{p_n}\equiv 1(mod p1p_1). Prove that a) (a−1)⋮pi(a-1)\vdots p_i for some i=1,..,ni=1,..,n b) Can (a−1)(a-1) be divisible by pip_i for exactly one ii of i=1,...,ni=1,...,n?
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