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Modulus congruence means that both numbers, 11 and 16 for example, have the same remainder after the same modular (mod 5 for example). 11 mod 5 has a remainder of 1. 11/5 = 2 R1. 16 mod 5 also has a remainder of 1. 16/5 = 3 R1. Therefore 11 and 16 are congruent through mod 5.
Mathematically, congruence modulo $n$ is an equivalence relation. We define: $$a\equiv b \pmod n \iff n\mid (a - b)$$ Equivalently: When working in $\pmod n$, any number $a$ is congruent $\mod n$ to an integer $b$ if there exists an integer $k$ for which $\;nk = (a - b)$.
Congruence Modulo is an Equivalence Relation. Convince yourself that the slices used in the previous example have the following properties: Every pair of values in a slice are related to each other. We will never find a value in more than one slice (slices are mutually disjoint)
Congruence. Given an integer m ≥ 1, called a modulus, two integers a and b are said to be congruent modulo m, if m is a divisor of their difference; that is, if there is an integer k such that a − b = k m.
Quick Reference. (modulo n) For each positive integer n, the relation of congruence between integers is defined as follows: a is congruent to b modulo n if a − b is a multiple of n. This is written a ≡ b (mod n ). The integer n is the modulus of the congruence.
Definition 5.3.1. Let a, b ∈ Z and . n ∈ N. We say that a is congruent to b modulo n when . n ∣ ( a − b). The “n” is refered to as the modulus and we write the congruence as . a ≡ b ( mod n). 🔗. When n ∤ ( a − b) we say that a is not congruent to b modulo , n, and write . a ≢ b ( mod n). 🔗.
An important equivalence is congruence modulo m of integers. We say x ≡ y(mod m) for integers x, y, m if there exists an integer h such that x − y = hm, that is, if m divides x − y. If x − y = hm and y − z = km, then x − z = x − y + y − z = hm + km.
