4. Masha chose a natural number $n$ and wrote down all natural numbers from 1 to 6 n on the board. Then, Masha halved half of these numbers, reduced a third of the numbers by a factor of three, and increased all the remaining numbers by a factor of six. Could the sum of all the resulting numbers match the sum of the original numbers? (20 points)
Solution: Yes, this could have happened. Let $n=2$, then the sum of all numbers from 1 to 12 is

$$
1+2+3+\ldots+12=78
$$

Suppose Masha halved the numbers 2, 3, 5, 9, 10, 11, reduced the numbers 4, 6, 8, 12 by a factor of three, and increased the numbers 1, 7 by a factor of six. Then the sum of the resulting numbers will be

$$
\frac{2+3+5+9+10+11}{2}+\frac{4+6+8+12}{3}+6(1+7)=78
$$

Answer: could have