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L(n),T(n)

Source: Iranian National Olympiad (3rd Round) 2006

August 26, 2006
abstract algebragroup theorynumber theory proposednumber theory

Problem Statement

For each nn, define L(n)L(n) to be the number of natural numbers 1≤a≤n1\leq a\leq n such that n∣an−1n\mid a^{n}-1. If p1,p2,…,pkp_{1},p_{2},\ldots,p_{k} are the prime divisors of nn, define T(n)T(n) as (p1−1)(p2−1)⋯(pk−1)(p_{1}-1)(p_{2}-1)\cdots(p_{k}-1). a) Prove that for each n∈Nn\in\mathbb N we have n∣L(n)T(n)n\mid L(n)T(n). b) Prove that if gcd⁡(n,T(n))=1\gcd(n,T(n))=1 then φ(n)∣L(n)T(n)\varphi(n) | L(n)T(n).
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