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Sequence of positive integers

Source: Chinese TST 2009 6th P3

April 4, 2009
inductionnumber theoryprime numbersrelatively primeprime factorizationgreatest common divisorMobius function

Problem Statement

Let (an)n≥1 (a_{n})_{n\ge 1} be a sequence of positive integers satisfying (am,an)=a(m,n) (a_{m},a_{n}) = a_{(m,n)} (for all m,n∈N+ m,n\in N^ +). Prove that for any n∈N+,∏d∣nadμ(nd) n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}} is an integer. where d∣n d|n denotes d d take all positive divisors of n. n. Function μ(n) \mu (n) is defined as follows: if n n can be divided by square of certain prime number, then μ(1)=1;μ(n)=0 \mu (1) = 1;\mu (n) = 0; if n n can be expressed as product of k k different prime numbers, then μ(n)=(−1)k. \mu (n) = ( - 1)^k.
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