2. How many five-digit numbers exist that are not divisible by 1000, and have the first, third, and last digits even?

Otvet: 9960.
Solution. The first digit of the number can be any of the four (2, 4, 6, or 8), the second and fourth can be any of ten each, and the third and fifth, if we abandon the condition "not divisible by a thousand," can be any of five (0, 2, 4, 6, or 8). Therefore, there are $4 \times 10 \times 5 \times 10 \times 5=10000$ five-digit numbers in which the first, third, and last digits are even; since among them $4 \times 10 \times 1 \times 1 \times 1=40$ numbers are divisible by 1000, the numbers that satisfy the condition of the problem will be $10000-40=9960$.

Criteria. If the solution is incorrect - 0 points.

Only the correct answer - 0 points.

If the reasoning is correct and there is a computational error - 3 points. If the solution is correct - 7 points.