Prove that if we can write the natural number $n$ as a sum of two squares then we can also write $k^2n$ as a sum of two squares.
We're given that we can write $n$ as the sum of two squares. Let these be $x^2$ and $y^2$.
$$ \begin{align} n & = x^2 + y^2 \\ \\ k^2n & = k^2(x^2 + y^2) \\ \\ & = (kx)^2 + (ky)^2 \end{align}$$
And so $k^2n$ can also be written as the sum of two squares.