  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.
