  We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of
integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number
$m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k
\pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary
methods, we answer fully this question for exponents $k \leq 10$, and we
further discuss some intermediary cases such as categorizing the values of $n$
such that every element in $\mathbb{Z}_n$ can be written as a sum of three
squares. Hensel's Theorem for $p$-adic integers plays a key role. Finally, we
give an application of this problem to the Erd\H os-Falconer distance problem
for rings $\mathbb{Z}_n^d$.
