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The Problem Solver's Resource

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Modulos

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

Occasionally, if two equivalent expressions are both modulated by the same number, the entire equation will be followed by the modulo.

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

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

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

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

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 $p$ , $(p-1)! \equiv -1 \pmod p$ .