Question 77
For each positive integer $n$, define $f(n)$ by
$$
f(n) = \begin{cases}
n^{100} & \text{(if digit sum of $n$ is even)}, \\
-n^{100} & \text{(if digit sum of $n$ is odd)}.
\end{cases}
$$
Let $S = f(1) + f(2) + \dots + f(10^{100} - 1)$, and determine the largest non-negative integer $m$ such that $S$ is divisible by $5^m$. (You can use the fact without proof that $S$ is nonzero.)
$$
f(n) = \begin{cases}
n^{100} & \text{(if digit sum of $n$ is even)}, \\
-n^{100} & \text{(if digit sum of $n$ is odd)}.
\end{cases}
$$
Let $S = f(1) + f(2) + \dots + f(10^{100} - 1)$, and determine the largest non-negative integer $m$ such that $S$ is divisible by $5^m$. (You can use the fact without proof that $S$ is nonzero.)