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Let $q$ be a prime. We give an elementary proof of the fact that for any$k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ thatcontain a $q^{th}$ power modulo almost every prime, is zero. This result holdsregardless of whether the proportion is measured additively ormultiplicatively. More specifically, the number of $k$-element subsets of $[-N,N]\cap\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime is nolarger than $a_{q,k} N^{k-(1-\frac{1}{q})}$, for some positive constant$a_{q,k}$. Furthermore, the number of $k$-element subsets of $\{\pmp_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{N}^{e_{N}} : 0 \leq e_{1}, e_{2}, \ldots,e_{N}\leq N\}$ that contain a $q^{th}$ power modulo almost every prime is nolarger than $m_{q,k} \frac{N^{Nk}}{q^{N}}$ for some positive constant$m_{q,k}$.

Almost No Finite Subset of Integers Contains a $q^{th}$ Power Modulo Almost Every Prime