Modular Arithmetic
Modular (often also
Modulo
) Arithmetic is an unusually versatile tool discovered by K.F.Gauss
(17771855) 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,..., [N1]
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,...,N1}:
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,...,N1} becomes a
ring
with
commutative
addition and multiplication. Division can't be always defined. To give an obvious
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 Spring '10
 Coluos
 Row, multiplication tables

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