**Proposition**: Complete and Reduced Residue Systems (Revised)

Let $a > 0$ and $b > 0$ be positive integers which are co-prime $a\perp b.$ Then the integer $ax+by$ runs through all values of

- a complete residue system modulo $ab,$ if the integers $x$ (respectively $y$) run through all values of the complete residue systems modulo $a$ (respectively $b,$)
- a reduced residue system modulo $ab,$ if the integers $x$ (respectively $y$) run through all values of the reduced residue systems modulo $a$ (respectively $b.$)

| | | | | created: 2019-04-29 17:21:20 | modified: 2019-04-29 17:31:07 | by: *bookofproofs* | references: [1272], [8152]

## 1.**Proof**: *(related to "Complete and Reduced Residue Systems (Revised)")*

(none)

[8152] **Jones G., Jones M.**: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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