An analogue of the Euclidean algorithm for square matrices of size 2 with integral non-negative entries and strictly positive determinant n defines a finite set R(n) of Euclid-reduced matrices corresponding to elements of {(a, b, c, d) ∈ N 4 | n = ab − cd, 0 ≤ c, d < a, b}. With Popeye's help 2 on the use of sails of lattices we show that R(n) contains d|n, d 2 ≥n d + 1 − n d elements.