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In 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 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