9. Let m and a be integers such that m≥ 1 and (a,m) = 1. Provethat if {r₁, ..., p(m)} is a reduced set of residues modulo m, then{ar₁,..., arp(m)} is also a reduced set of residues modulo m.

Answered: 9. Let m and a be integers such that m…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

9. Let m and a be integers such that m≥ 1 and (a,m) = 1. Prove
that if {r₁, ..., p(m)} is a reduced set of residues modulo m, then
{ar₁,..., arp(m)} is also a reduced set of residues modulo m.

Expert Solution

Step 1: Explanation

Consider that $a subscript 1 comma a subscript 2. a subscript 3..... a subscript n$ is a complete set of residues modulo m.

It is given that gcd ( a,m) = 1

The objective is to claim that $a a subscript 1 comma a a subscript 2 comma a a subscript 3.... comma a a subscript n$ is a complete set of residues modulo m

We know that any set of integers form a complete set of residues modulo m if and only if all the integers are congruent to different integer modulo m.

That is $a subscript 1 comma a subscript 2 comma.... a subscript n$ is a complete set of residues modulo m.Then, a_{i $\neq$}a_j (mod m) for every

1 $\leq$ i $\leq$ j $\leq$ m.

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9. Let m and a be integers such that m > 1 and (a,m)= 1. Prove that if {r1, ...,r(m)} is a reduced set of residues modulo m, then {arı,..., arp(m)} is also a reduced set of residues modulo m.