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Proposition: Creation of Complete Residue Systems From Others

Let $m > 0$ be a positive integer and let $n\perp m$ be co-prime. If $C=\{a_1,\ldots,a_m\}$ is a complete residue system modulo $m$, then $nC:=\{na_1,\ldots,na_m\}$ is also a complete residue system modulo $m.$

Example

We can choose $n\cdot 0,n\cdot 1,\ldots, n\cdot(m-1)$ for any $n$ with $\gcd(n,m)=1$ as a complete residue system modulo $m.$

Table of Contents

Proofs: 1

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Proofs: 1 2 3

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998