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product of k<n such that gcd(k,n)=1

Source: Mongolian MO 2007 Grade 11 P2

April 8, 2021
number theory

Problem Statement

For all n≥2n\ge2, let ana_n be the product of all coprime natural numbers less than nn. Prove that
(a) n∣an+1⇔n=2,4,pα,2pαn\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha (b) n∣an−1⇔n≠2,4,pα,2pαn\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha
Here pp is an odd prime number and α∈N\alpha\in\mathbb N.
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