Snake graphs and continued fractions

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.