  We observe structure in the sequences of quotients and remainders of the
Euclidean algorithm with two families of inputs. Analyzing the remainders, we
obtain new algorithms for computing modular inverses and representating prime
numbers by the binary quadratic form $x^2 + 3xy + y^2$. The Euclidean algorithm
is commenced with inputs from one of the families, and the first remainder less
than a predetermined size produces the modular inverse or representation.
