## Task B-1.4.

We add the square of an integer $a \neq 1$ to the integer itself and subtract one, then divide the resulting number by $a-1$. For which integer $a$ is the quotient obtained in this way an integer?
## Solution.

So, we need to determine all integers $a$ for which the fraction $\frac{a+a^{2}-1}{a-1}$ is an integer.

This fraction can be written in the following form:

$$
\begin{gathered}
\frac{a+a^{2}-1}{a-1}=\frac{a^{2}-1+a}{a-1}=\frac{(a-1)(a+1)+(a-1)+1}{a-1}= \\
\frac{(a-1)(a+1)}{a-1}+\frac{a-1}{a-1}+\frac{1}{a-1}=a+1+1+\frac{1}{a-1}=a+2+\frac{1}{a-1}
\end{gathered}
$$

It is now clear that the fraction $\frac{a+a^{2}-1}{a-1}$ will be an integer only if the fraction $\frac{1}{a-1}$ is an integer, i.e., if $a-1$ is equal to -1 or 1. Therefore, $a \in\{0,2\}$.

Note: If a student guesses the solutions, they get one point for each correct solution.