Problem 2. In a two-digit number, each digit was increased by 2 or by 4 (different digits could be increased by different numbers), as a result of which the number increased fourfold. What could the original number have been? Find all possible options and prove that there are no others.
Solution. Answer: 14.

Let $x$ be the original number, then the resulting number equals $4 x$. In this case, after increasing two digits, the number itself was increased by 22, 24, 42, or 44. This results in four cases:

- $4 x-x=22$
- $4 x-x=24$
- $4 x-x=42$
- $4 x-x=44$.

Among them, only the third one fits, corresponding to the value $x=14$.

One highest suitable criterion is used:

7 points. Any complete solution to the problem.

3 points. Only the correct answer.

If the problem is reduced to considering the four cases mentioned in the solution, then 1 point is deducted for each case not correctly analyzed.