We develop a formula for the number of non-negative integer solutions (x,y,z) of the equation ax + by + cz = n, where a, b, c, and n are given positive integers. The formula leads us to a surprising connection between the number of non-negative integer solutions of the equation ax + by + cz = n and quadratic residues. As a consequence of our work, we are able to prove the equivalence between two fundamental results by Gauss and Sylvester in the nineteenth century that are generally viewed as independent.