{- |
Description: Modular multiplicative inverse
Maintainer: dev@chungyc.org

Part of Ninety-Nine Haskell "Problems".  Some solutions are in "Solutions.P42".
-}
module Problems.P42 (multiplicativeInverse) where

import qualified Solutions.P42 as Solution

{- |
In [modular arithmetic](https://brilliant.org/wiki/modular-arithmetic/),
integers $$a$$ and $$b$$ being congruent modulo an integer $$n$$,
also written as $$a \equiv b \pmod{n}$$, means that $$a - b = kn$$ for some integer $$k$$.
Many of the usual rules for addition, subtraction, and multiplication in ordinary arithmetic
also hold for modular arithmetic.

A multiplicative inverse of an integer $$a$$ modulo $$n$$ is an integer $$x$$ such that $$ax \equiv 1 \pmod{n}$$.
It exists if and only if $$a$$ and $$n$$ are coprime.

Write a function to compute the multiplicative inverse $$x$$ of a given integer $$a$$ and modulus $$n$$
lying in the range $$0 \leq x < n$$.
Use the [extended Euclidean algorithm](https://brilliant.org/wiki/extended-euclidean-algorithm/)
and the fact that $$ax \equiv 1 \pmod{n}$$ if $$ax+ny=1$$.

=== Examples

>>> multiplicativeInverse 3 5
Just 2

>>> multiplicativeInverse 48 127
Just 45

>>> multiplicativeInverse 824 93
Just 50

>>> multiplicativeInverse 48 93
Nothing
-}
multiplicativeInverse :: Integral a => a -> a -> Maybe a
multiplicativeInverse :: forall a. Integral a => a -> a -> Maybe a
multiplicativeInverse = a -> a -> Maybe a
forall a. Integral a => a -> a -> Maybe a
Solution.multiplicativeInverse