1. Determine the unknown digit $a$ of the number $\overline{401512 a}$ so that the remainders when this number is divided by 3 and 5 are equal.
First method:

Possible remainders when dividing by 3 are 0, 1, and 2.

If the remainder when dividing by 3 is 0, then the number $\overline{401512 a}$ is divisible by 3, and the sum of its digits must be divisible by 3.

The sum of the digits of the number $\overline{401512 a}$ is $13+a$, so the digit a can be 2, 5, and 8. The numbers of the given form that are divisible by 3 are 4015122, 4015125, and 4015128. Of these three numbers, only the number 4015125 is divisible by 5.

If the remainder when dividing by 3 is 1, then the digit a can be 0, 3, 6, and 9. The numbers of the given form that have a remainder of 1 when divided by 3 are 4015120, 4015123, 4015126, and 4015129.

Of these four numbers, only the number 4015126 has a remainder of 1 when divided by 5.

If the remainder when dividing by 3 is 2, then the digit a can be 1, 4, and 7. The numbers of the given form that have a remainder of 2 when divided by 3 are 4015121, 4015124, and 4015127.

Of these three numbers, only the number 4015127 has a remainder of 2 when divided by 5.