  In a recent work, the present author generalized a fundamental result of
Gauss related to quadratic reciprocity, and also showed that the above result
of Gauss is equivalent to a special case of a well-known result of Sylvester
related to the Frobenius coin problem. In this note, we use this equivalence to
show that the above generalization of the result of Gauss naturally leads to an
interesting generalization of the result of Sylvester. To be precise, for given
positive coprime integers $a$ and $b$, and for a family of values of $k$ in the
interval $0 \leq k < (a-1)(b-1)$, we find the number of nonnegative integers
$\leq k$ that can be expressed in the form $ax+by$ for nonnegative integers $x$
and $y$. We also give an elementary proof of Eisenstein's Lemma for Jacobi
symbols using floor function sums. Our proof provides a natural straightforward
generalization of the Gauss-Eisenstein proof of the law of quadratic
reciprocity for Jacobi symbols.
