Difference between revisions of "Modular arithmetic"
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=== Applications === | === Applications === | ||
− | Modular arithmetic is an extremely useful tool in mathematics competitions. It enables us to easily solve [[Linear | + | Modular arithmetic is an extremely useful tool in mathematics competitions. It enables us to easily solve [[Linear Diophantine equation]]s, and it often helps with other [[Diophantine equation | Diophantine equations]] as well. |
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== Intermediate == | == Intermediate == |
Revision as of 21:53, 23 June 2006
Modular arithmetic is a special type of arithmetic that involves only integers. Given integers ,
, and
, with
, we say that
is congruent to
modulo
, or
(mod
), if the difference
is divisible by
.
For a given positive integer , the relation
(mod
) is an equivalence relation on the set of integers. This relation gives rise to an algebraic structure called the integers modulo
(usually known as "the integers mod
," or
for short). This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to Diophantine equations, testing whether certain large numbers are prime, and even some problems in cryptology.
Contents
Introductory
Useful Facts
Consider four integers and a positive integer
such that
and
. In modular arithmetic, the following identities hold:
- Addition:
.
- Substraction:
.
- Multiplication:
.
- Division:
, where
is a positive integer that divides
and
.
- Exponentiation:
where
is a positive integer.
Examples
Applications
Modular arithmetic is an extremely useful tool in mathematics competitions. It enables us to easily solve Linear Diophantine equations, and it often helps with other Diophantine equations as well.