Answer:
15
Explanation:
In order to compare $n\sqrt{2}$ to $20$, we can compare the square of $n\sqrt{2}$ to the square of $20.$ We have
\begin{align*}
\left(n\sqrt{2}\right)^2 &= \left(n\sqrt{2}\right)\left(n\sqrt{2}\right) = n^2 \left(\sqrt{2}\right)^2 = n^2\cdot 2= 2n^2,\\
20^2 &= 400.
\end{align*}Therefore, we have $n\sqrt{2} > 20$ whenever $n^2 > 200.$ Since $14^2 = 196$ and $15^2 = 225,$ we know that $\boxed{15}$ is the smallest integer $n$ such that $n\sqrt{2}$ is greater than $20.$