## Task B-4.1.

For which natural numbers $n$ will the expression $\left(\frac{3+i}{2-i}\right)^{n}$ be a real number? How many such numbers $n$ are less than 2016?
## Solution.

$$
\left(\frac{3+i}{2-i}\right)^{n}=\left(\frac{3+i}{2-i} \cdot \frac{2+i}{2+i}\right)^{n}=(1+i)^{n}
$$

Consider two cases.

First case for $n=2 k$:

$(1+i)^{n}=(1+i)^{2 k}=(2 i)^{k}$.

This will be a real number only if $k$ is even, i.e., $n=4 l$.

Second case for $n=2 k+1$:

$(1+i)^{n}=(1+i)^{2 k+1}=(2 i)^{k} \cdot(1+i)$.

This expression is never a real number.

1 point

The number of natural numbers less than 2016 that are divisible by 4 is 503.