Modular arithmetic

Modular arithmetic a special type of arithmetic that involves only integers. If two integers ${a},{b}$ leave the same remainder when they are divided by some positive integer ${m}$ , we say that ${a}$ and $b$ are congruent modulo ${m}$ or $a\equiv b \pmod {m}$ .

Introductory

Operations

Consider four integers ${a},{b},{c},{d}$ and a positive integer ${m}$ such that $a\equiv b\pmod {m}$ and $c\equiv d\pmod {m}$ . In modular arithmetic, the following operations are allowed:

Addition: $a+c\equiv b+d\pmod {m}$ .
Substraction: $a-c\equiv b-d\pmod {m}$ .
Multiplication: $ac\equiv bd\pmod {m}$ .
Division: $\frac{a}{e}\equiv \frac{b}{e}\pmod {\frac{m}{\gcd(m,e)}}$ , where $e$ is a positive integer that divides ${a}$ and $b$ .
Exponentiation: $a^e\equiv b^e\pmod {m}$ where $e$ is a positive integer.

Examples

${7}\equiv {1} \pmod {2}$

$49^2\equiv 7^4\equiv (1)^4\equiv 1 \pmod {6}$

$7a\equiv 14\pmod {49}\implies a\equiv 2\pmod {7}$

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.

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Modular arithmetic

Contents

Introductory

Operations

Examples

Applications

Intermediate

Topics

See also