The number m is called the modulus of the congruence. Congruence modulo m divides the The following basic properties follow from the.
We read this as “a is congruent to b modulo (or mod) n. For example 29 ? 8 mod 7
[a]n is the set of all integers that are congruent to a modulo n; i.e.
We say that a is not congruent to b modulo m and write a ? b (mod m)
Definition and Basic Properties. Definition 2.1.1. Let m be a positive integer. For a b ? Z we say that a is congruent to b modulo m and write a ? b (mod
It is to be note that any two integer are congruence modulo 1 whereas two integers are congruence modulo. 2 they are both even or both odd. Inasmuch as
If a is not congruent to b modulo m we write a ? b( mod m). Arithmetic modulo m. The operations +m and ·m satisfy many of the same properties as.
9 May 2011 CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE. POD FUNCTION. SILVIU RADU AND JAMES A. SELLERS. Abstract. In this paper we prove arithmetic ...
Congruences modulo powers of 13 have been considered by Atkin and O'Brien [2]. A general treatment of Pkiv) modulo powers of.
BASIC PROPERTIES OF CONGRUENCES The letters a;b;c;d;k represent integers The letters m;n represent positive integers The notation a b (mod m) means that m divides a b We then say that a is congruent to b modulo m 1 (Re exive Property): a a (mod m) 2 (Symmetric Property): If a b (mod m) then b a (mod m) 3
Congruences are an important and useful tool for the study of divisibility As we shall see they are also critical in the art of cryptography De nition 3 1If a and b are integers and n>0wewrite a bmodn to mean nj(b ?a) We read this as a is congruent to b modulo (or mod) n For example 29 8 mod 7 and 60 0 mod 15
Congruence mod n has the following two properties (1) If a 2Z and r is the remainder when a is divided by n then [a] = [r] (2) There are exactly n distinct congruence classes modulo n namely [0];[1];:::;[n 1] Remark The set f[0];[1];:::;[n 1]gis denoted by Z n
Congruences: We say a is congruent to b modulo m and write a b mod m if a and b have the same remainder when divided by m or equivalently if a b is divisible by m Equivalently the congruence notation a b mod m can be thought of as a shorthand notation for the statement there exists an integer k such that a = b+ km "
Let n ? N Theorem 2 tells us that there are exactly n congruence classes modulo n A set containing exactly one integer from each congruence class is called a complete system of residues modulo n Examples The set {012 n ?1} of remainders is a complete system of residues modulo n by Theorem 2
By the de?nition of congruence modulo m this is the same as saying that a+c is congruent to b+d modulo msincea+c and b+d di?er by an integer multiple (j +k) of m In symbols we have: a+c ? b+d (mod m) (68) as desired A similar proof can be used to show that if a ? b (mod m) and c ? d (mod m) then ac ? bd (mod m)