Modular Arithmetic

Modular Arithmetic - Modular Arithmetic Modular(often also...

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Modular Arithmetic Modular (often also Modulo ) Arithmetic is an unusually versatile tool discovered by K.F.Gauss (1777-1855) in 1801. Two numbers a and b are said to be equal or congruent modulo N iff N|(a- b), i.e. iff their difference is exactly divisible by N. Usually (and on this page) a,b, are nonnegative and N a positive integer. We write a = b (mod N). The set of numbers congruent to a modulo N is denoted [a] N . If b [a] N then, by definition, N|(a- b) or, in other words, a and b have the same remainder of division by N. Since there are exactly N possible remainders of division by N, there are exactly N different sets [a] N . Quite often these N sets are simply identified with the corresponding remainders: [0] N = 0, [1] N = 1,..., [N-1] N = N- 1. Remainders are often called residues ; accordingly, [a]'s are also know as the residue classes . It's easy to see that if a = b (mod N) and c = d (mod N) then (a+c) = (b+d) (mod N). The same is true for multiplication. These allows us to introduce an algebraic structure into the set {[a] N : a=0,1,...,N-1}: By definition, 1. [a] N + [b] N = [a + b] N 2. [a] N × [b] N = [a × b] N Subtraction is defined in an analogous manner [a] N - [b] N = [a - b] N and it can be verified that thus equipped set {[a] N : a=0,1,...,N-1} becomes a ring with commutative addition and multiplication. Division can't be always defined. To give an obvious
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