Question 81
Let $k \in \mathbb{Z}$ prove that there are infinitely many pairs of distinct positive integer numbers $n, m$ such that
$$
\begin{aligned}
n + S(2n) &= m + S(2m), \\
kn + S(n^2) &= km + S(m^2),
\end{aligned}
$$
where $S(n)$ is the sum of the digits of $n$ to base 10.
$$
\begin{aligned}
n + S(2n) &= m + S(2m), \\
kn + S(n^2) &= km + S(m^2),
\end{aligned}
$$
where $S(n)$ is the sum of the digits of $n$ to base 10.