Computing mod inverse? How might one compute $4^{-1} \mod 17$ I know the answer is 13. I'm just not sure how to arrive at that number, and can't find any good explanations. Any help would be great
In order to compute the inverse of $a$ modulo $n$, use the extended Euclidean algorithm to find the GCD of $a$ and $n$ (which should be 1), together with coefficients $x,y$ such that $ax + ny = 1$. The inverse of $a$ modulo $n$ is thus $x$.

The extended Euclidean algorithm gives a constructive proof of Bézout's identity, which states that for all integers $a,b$ there exist integers $x,y$ such that $ax+by = \mathrm{gcd}(a,b)$. A different proof shows that the minimal positive value of $ax+by$ (over all $x,y$) is $\mathrm{gcd}(a,b)$.

The extended Euclidean algorithm works in greater generality, for any Euclidean domain. An important example is the ring of polynomials over a field.