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The Problem Solver's Resource

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Modulos

Definition

$n\equiv a\pmod{b}$ if $n$ is the remainder when $a$ is divided by $b$ to give an integral amount.

Special Notation

Properties

For any number there will be only one congruent number modulo $m$ between $0$ and $m-1$ .

If $a\equiv b \pmod{m}$ and $c \equiv d \pmod{m}$ , then $(a+c) \equiv (b+d) \pmod {m}$ .

$a \pmod{m} + b \pmod{m} \equiv (a + b) \pmod{m}$

$a \pmod{m} - b \pmod{m} \equiv (a - b) \pmod{m}$

$a \pmod{m} \cdot b \pmod{m} \equiv (a \cdot b) \pmod{m}$

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Useful Theorems

Fermat's Little Theorem:For a prime $p$ and a number $a$ such that $a\ne{p}$ , $a^{p-1}\equiv 1 (\pmod{p}$ . Wilson's Theorem: For a prime $p$ , $(p-1)! \equiv -1 (mod p)$ .

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