MathDB

n|a if sum 1/a_(\phi (k)})=a/b

Source: Mathcenter Contest / Oly - Thai Forum 2012 (R1) p4 sl-12 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 13, 2022

number theory

Problem Statement

Given a natural n>2, let

\{ a_1,a_2,...,a_{\phi (n)} \} \subset \mathbb{Z}

is the Reduced Residue System (RRS) set of modulo

n

(also known as the set of integers

k

where

(k,n)=1

and no pairs are congruent in modulo

n

). if write

\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{\phi (n)}}=\frac{a}{b}

where

a,b \in \mathbb{N}

and

(a,b)=1

, then prove that

n|a

.
(PP-nine)