Modular multiplicative inverse

In mathematics, in particular, the area of number theory, a modular multiplicative inverse of an integer $a$ is an integer $x$ such that the product $ax$ is congruent to 1 with respect to the modulus $m$ . In the standard notation of modular arithmetic this congruence is written as

which is the shorthand way of writing the statement that $m$ divides (evenly) the quantity $ax - 1$ , or, put another way, the remainder after dividing $ax$ by the integer $m$ is 1. If $a$ does have an inverse modulo $m$ there are an infinite number of solutions of this congruence which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to $a$ (i.e., in $a$ 's congruence class) will have any element of $x$ 's congruence class as a modular multiplicative inverse. Using the notation of ${\overline {w}}$ to indicate the congruence class containing $w$ , this can be expressed by saying that the modulo multiplicative inverse of the congruence class ${\overline {a}}$ is the congruence class ${\overline {x}}$ such that:

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