Answer: 223332.

Sketch of the solution. If a number is divisible by 3, then the sum of its digits is divisible by 3, which means the number of twos is a multiple of three, and thus, there must be at least three twos. Therefore, the desired number is a six-digit number. Since it is divisible by 2, it ends in 2. The number is smaller the smaller the leading digits are, hence the answer.

Criteria. Correct answer without justification: 2 points.
1. Find the smallest number that contains only the digits 2 and 3 in equal quantities, and is divisible by 2 and 3.