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Congruence Classes and Modular Arithmetic

Congruence classes (or shorter congruences) are powerful mathematical tools, simplifying many calculations with integers. The arithmetic associated with congruences is called modular arithmetic. In this kind of arithmetic, we simplify calculations by replacing each integer with its remainder when divided by some fixed positive integer $n.$ The simplification effect of this is that the whole, infinite set of integers $\mathbb Z$ is replaced by a much smaller set $\mathbb Z_n$ which contains only a finite number $n$ of elements. We will see that we can add, subtract and multiply numbers in $\mathbb Z_n.$ Just as it is the case in $\mathbb Z$, there are some difficulties with the division. In this sense, $\mathbb Z_n$ inherits many properties of integers but, because it is finite, it is much easier to work with.

| | | | created: 2014-02-20 21:52:51 | modified: 2019-04-10 20:12:07 | by: bookofproofs | references: [1272], [8152]

1.Definition: Congruent, Residue

2.Proposition: Congruence Classes

3.Proposition: Congruences and Division with Quotient and Remainder

4.Explanation: Explanation of Congruence Classes

5.Proposition: Addition, Subtraction and Multiplication of Congruences

6.Proposition: Cancellation of Congruences With Factor Co-Prime To Module

7.Proposition: Cancellation of Congruences with General Factor

8.Proposition: Multiplication of Congruences with a Positive Factor

9.Proposition: Congruence Modulo a Divisor

10.Diophantine Equations and Complete and Reduced Residue Systems


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Bibliography (further reading)

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

[1272] Landau, Edmund: “Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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