1. The number $\overline{6 x 62 y 4}$ is divisible by 11, and when divided by 9, it gives a remainder of 6. Find the remainder when this number is divided by $13 .(15$ points)
# Solution

By the divisibility rule for 11, we get

$((x+2+4)-(6+6+y)) \vdots 11$ or $(x-6-y) \vdots 11$

Let's find suitable options: $(0 ; 5)(1 ; 6)(2 ; 7)(3 ; 8)(4 ; 9)(6 ; 0)(7 ; 1)(8 ; 2)$ $(9 ; 3)$.

If the number $\overline{6 x 62 y 4}$ gives a remainder of 6 when divided by 9, then by the divisibility rule for 9 we get

$$
\begin{aligned}
& (6+x+6+2+y+4)-6: 9 \\
& \text { or }(12+x+y) \vdots 9
\end{aligned}
$$

The combination $(6 ; 0)$ fits, so the original number is 666204.

The remainder when this number is divided by 13 is 6.

Answer: 6.