Solution: Let $ x \equiv 1 \pmod9 $ and $ x \equiv 1 \pmod11 $. This implies $ x \equiv 1 \pmod99 $, since 9 and 11 are coprime. The two-digit numbers satisfying this are $ 99 + 1 = 100 $ (not two-digit) and $ 1 $. However, the next smaller solution is $ 1 $, which is not two-digit. Thus, no two-digit number satisfies both conditions. - Groen Casting