What is the total number of digits used when the first 2002 positive even integers are written?
I notice that the first 2002 positive even integers are 2, 4, 6, ..., 4004. To count the digits, I can group them by how many digits they have. For example, there are 4 one-digit even numbers: 2, 4, 6, and 8. They use 4 digits in total. There are 45 two-digit even numbers: 10, 12, ..., 98. They use 45 x 2 = 90 digits in total. There are 450 three-digit even numbers: 100, 102, ..., 998. They use 450 x 3 = 1350 digits in total. The rest of the even numbers are four-digit numbers: 1000, 1002, ..., 4004. There are 2002 - 4 - 45 - 450 = 1503 of them. They use 1503 x 4 = 6012 digits in total. Therefore, the total number of digits used is 4 + 90 + 1350 + 6012 = 7456. 