Q:

A small question about multiplicative inverses

I wan't to find the multiplivative inverse of $[22]^{-1}_{12} $ in $Z_{12}$
But when I do the euclidean algorithm on 22 and 12 I get : gcd(22, 12) = 2
Does that mean there are no multiplicative inverses? I only know how to get the multiplicative inverse if gcd(22, 12) = 1.
Are there any other ways to get the multiplicative inverse of $[22]^{-1}_{12} $ in $Z_{12}$

A:

Suppose that there was a multiplicative inverse of $22$ in $\Bbb Z_{12}$, call it $k$.  Then $22k\equiv 1\pmod{12}$, implying that there exists an integer $n$ such that $22k=1+12n$, but this is an impossibility since the number on the left is even but the number on the right is odd for every integer values of $k$ and $n$
Therefore, $22$ does not have a multiplicative inverse in $\Bbb Z_{12}$.
In general, for $x$ to have a multiplicative inverse in $\Bbb Z_n$ one requires that $\gcd(x,n)=1$

