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Definition (Modular Multiplicative Inverse and $U(n)$)
Let $n$ be a positive integer, and let $a\in \mathbb{Z}_n$. Then the modular multiplicative inverse of $a \mod n$ is an integer $b\in \mathbb{Z}_n$ such that $(ab)\mod n=1$. If such an integer exists, then we'll write $a^{-1}\mod n$ to represent the modular multiplicative inverse. If the context is clear, we may just refer to this as the inverse instead of the modular multiplicative inverse. We'll use the notation $U(n)$ to represent the set of elements in $\mathbb{Z}$ that have a modular multiplicative inverse. Notationaly, we write $$U(n)=\{a\in\mathbb{Z}_n\mid \text{ there exists } b\in \mathbb{Z}_n \text{ such that } (ab)\mod n=1\}.$$
The Sage code "inverse_mod(a,n)" will compute the modular multiplicative inverse of $a\mod n$ or return an error.
inverse_mod(7, 11)
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