  We find a new approach to computing the remainder of a polynomial modulo
$x^n-1$; such a computation is called modular enumeration. Given a polynomial
with coefficients from a commutative $\mathbb{Q}$-algebra, our first main
result constructs the remainder simply from the coefficients of residues of the
polynomial modulo $\Phi_d(x)$ for each $d\mid n$. Since such residues can often
be found to have nice values, this simplifies a number of modular enumeration
problems; indeed in some cases, such residues are already known while the
related modular enumeration problem has remained unsolved. We list six such
cases which our technique makes easy to solve. Our second main result is a
formula for the unique polynomial $a$ such that $a \equiv f \mod \Phi_n(x)$ and
$a\equiv 0 \mod x^d-1$ for each proper divisor $d$ of $n$.
  We find a formula for remainders of $q$-multinomial coefficients and for
remainders of $q$-Catalan numbers modulo $q^n-1$, reducing each problem to a
finite number of cases for any fixed $n$. In the prior case, we solve an open
problem posed by Hartke and Radcliffe. In considering $q$-Catalan numbers
modulo $q^n-1$, we discover a cyclic group operation on certain lattice paths
which behaves predictably with regard to major index. We also make progress on
a problem in modular enumeration on subset sums posed by Kitchloo and Pachter.
