Algebra 5 Congruence Classes

Congruence classes Congruence modulo m is an equivalence relation on ℤ. Thus the equivalence classes [𝑎] 𝑚 form a partition of ℤ. [𝑎] 𝑚 is called the congruence class of a: 𝑎+(multiple of 𝑚)

Proposition 1 For 𝑎, 𝑎’ in ℤ, 𝑎≡ 𝑎 ′ (𝑚𝑜𝑑 𝑚) if and only if [𝑎] 𝑚 = [𝑎′] 𝑚

Proposition 2 Suppose [𝑎] 𝑚 and [𝑏] 𝑚 are two congruence classes and 𝑐’ in ℤ, is in both [𝑎] 𝑚 and [𝑎] 𝑚 . Then [𝑎] 𝑚 = [𝑏] 𝑚 .

ℤ/𝑚ℤ ℤ/𝑚ℤ= [0] 𝑚 , [1] 𝑚 , … [𝑚−1] 𝑚 .

Complete Sets of Representatives A complete set of representatives for ℤ/𝒎ℤ is a set of integers { 𝑟 1 ,… 𝑟 𝑚 } so that every integer is congruent modulo 𝑚 to exactly one of the numbers in the set. Thus {0, 1, 2,…𝑚−1} is a complete set of representatives for ℤ/𝑚ℤ .

Theorem 3 Primitive Root Theorem Let 𝑝 be a prime number. There exists some integer b so that {0,𝑏, 𝑏 2 , 𝑏 3 ,…, 𝑏 𝑝−1 } is a complete set of representatives for ℤ/𝑚ℤ .