## Task B-1.1.

The numbers 10101 and 13226 have the same remainder when divided by the same three-digit number. Determine that remainder.
## Solution.

Let $x$ be a three-digit number, and $r$ the desired remainder. Then the following equations hold.

$$
\begin{aligned}
& 10101=x \cdot m+r \\
& 13226=x \cdot n+r \\
& 1 \text { point }
\end{aligned}
$$

Then we have

$$
\begin{aligned}
& 13226-10101=x(n-m) \\
& \Rightarrow 3125=x(n-m)
\end{aligned}
$$

We conclude that $x$ is a three-digit divisor of the number $3125=5^{5}$. The only three-digit divisors of 3125 are powers of 5, the numbers 625 and 125. We calculate

$$
\begin{aligned}
& 13226=625 \cdot 21+101=125 \cdot 105+101 \\
& 10101=625 \cdot 16+101=125 \cdot 80+101
\end{aligned}
$$

Notice that in both cases the remainder is the same and equals 101.