Modular Arithmetic w/o Proofs

Fix a positive integer n, We are going to divide a system of numbers called modular numbers that can be added and multiplied according to precise rule. There are n number of these labeled as...

$$ \bar{0}, \bar{1}, \bar{2}, ... , \bar{n}-2, \bar{n} -1 $$

The collection of these numbers is denoted as $Z / N$ and, when given the addition and multiplicative operations, is known as the ring of integers modulo n.

$$ Z/N=({\bar{O}, \bar{1}, .., \bar{n}-1}), + ,x $$

Example: n = 4 $Z/4 = [\bar{0}, \bar{1}, \bar{2}, \bar{3}], + x$

+	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

The multiplication table would then be...

x	0	1	2	3
0	0	0	0	0
1	0	1	2	3
2	0	2	0	2
3	0	3	2	1

Things to notice

1. 0 bar is the additive identity: 0 + a = a
2. 0 * a = 0
3. 1 bar is the multiplicative identity: 1 * a = a
4. 3 bar has a multiplicative inverse, an element that gives 1 bar upon multiplication.
5. 2 bar does not have a multiplicative inverse but does have the property that there is a non-zero element that multiplies it to 0 bar: 2 * 2 = 0

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Some Definitions and Property's

1. a bar + b bar = (a + b) bar
2. a bar * b bar = ab bar
3. 0 bar is the additive identity
4. 1 bar is the multiplicative identity
5. if a bar * b bar = 1 then they are multiplicative inverses
6. if a bar doesn't equal 0, and b bar doesn't equal zero but a bar * b bar is equal to 0 then they are 0 divisors.

The r is the residue of a modulo n, and it is written as the following:

$$ r = a modn $$ So for example 15 mod 4 is equal to 3, since 15 = 4 * 3 + 3 Notice that we can expand upon this and derive more conclusions.

General Fact we need to prove

a bar is a unit of Z/N if and only if a and n are relatively prime (share no common prime factors) 12 = 2 * 2 * 3

0 ,1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

units: zero divisors: