The inverse of m with respect to n in modular arithmetics From concrete mathematics problem 4.35.
Let $I(m,n)$ be function that satisfies the relation
$$  I(m,n)m + I(n,m)n = \gcd(m,n),$$
when $m,n \in \mathbb{Z}^+$ with $m ≠ n$. Thus, $I(m,n) = m'$ and $I(n,m) = n'$ in (4.5).
The value of $I(m,n)$ is an inverse of $m$ with respect to $n$. Find a recurrence that defines $I(m,n)$.
The (4.5) is just $m'm +n'n = \gcd(m,n)$.
What is meant by "The value of $I(m,n)$ is an inverse of $m$ with respect to $n$"? This tells us what relationship among these three values?

If $\gcd(m,n)=1$, so that the equation is $I(m,n)m+I(n,m)n=1$, then $I(m,n)$ is the multiplicative inverse of $m$ modulo $n$, since looking at that equation modulo $n$ yields $I(m,n)m\equiv1\pmod n$.
When $\gcd(m,n)>1$, there is no multiplicative inverse of $m$ modulo $n$, but $I(m,n)$ would be a multiplicative inverse of $m/\gcd(m,n)$ modulo $n/\gcd(m,n)$. Maybe that's good enough to warrant the phrase "inverse of $m$ with respect to $n$", although it's not standard as far as I'm concerned.
