1. Variant 1.

When multiplying a two-digit and a three-digit number, a four-digit number of the form $A=\overline{a b a b}$ is obtained. Find the largest $A$, given that $A$ is divisible by 14.
Answer: 9898.

Solution.

Note that $A=\overline{a b a b}=\overline{a b} \cdot 101$. Since 101 and 14 are coprime, $\overline{a b}$ is divisible by 14. The maximum value of $\overline{a b}=98$.