Task B-2.1. How many natural numbers $n<2013$ are there for which

$$
\left(i^{n}+i^{n+1}+i^{n+2}+\ldots+i^{n+2012}\right)^{2013}
$$

is a natural number?
## Solution.

Let's simplify the given expression.

$$
\begin{aligned}
\left(i^{n}+i^{n+1}+i^{n+2}+\ldots+i^{n+2012}\right)^{2013} & =i^{2013 n}\left(i^{0}+i^{1}+i^{2}+i^{3}+\ldots+i^{2012}\right)^{2013} \\
& =i^{2013 n}(1+i-1-i+\ldots+1)^{2013} \\
& =i^{2013 n}(1)^{2013}=i^{n}
\end{aligned}
$$

$i^{n}$ is a natural number if $n$ is divisible by four, i.e., of the form $n=4k$,