elementary number theory - Given two coprime integers, find multiples of them that differ by 1

I'll describe the problem with an example.

Find integers $n$ and $m$ such that $n\cdot13 + m\cdot101 = 1$.

This is the same as solving the equation $n\cdot13 \equiv 1 \pmod {101}$

I'm revising for a maths exam in a few days and it seems like a lot of the questions and examples rely on an ability to solve this type of problem. I can't seem to find a method in my notes, the solutions just get plucked out of thin air!

What is a neat way of doing it on pen and paper with a calculator?