Lockers are numbered with consecutive positive integers beginning with 1, and the digit 2 is used exactly 106 times (including on the last locker). What is the number of the last locker?

250 Verified answer:250

250 To find the number of the last locker, we need to determine the total number of lockers in which the digit 2 is used.

Considering that the lockers are numbered consecutively, we can observe that the digit 2 is used in the following number of lockers: 2, 12, 20-29, 32, 42, 52, ..., 92, 102, 112, ..., 192, 202, 212, ..., 902, 912, 920-929, totaling 106 occurrences.

Now, we can break down the ranges of locker numbers where the digit 2 is used:

1. Ranges from 2 to 9: In each of these ranges, the digit 2 is used once. So, we have 9 - 2 + 1 = 8 lockers.

2. Ranges from 20 to 29, 120 to 129, ..., 920 to 929: In each of these ranges, there are ten lockers where the digit 2 appears at the first position, thus 10 lockers for each range. Since there are 8 ranges, we have a total of 8 x 10 = 80 lockers.

3. Ranges from 200 to 299, 1200 to 1299, ..., 9200 to 9299: In each of these ranges, there are 100 lockers where the digit 2 appears at the second position, thus 100 lockers for each range. Since there are 8 ranges, we have a total of 8 x 100 = 800 lockers.

By adding up the number of lockers from these three cases, we get 8 + 80 + 800 = 888 lockers in which the digit 2 is used.

Since we know that the digit 2 is used exactly 106 times, we can subtract those lockers from the total count: 106 - 888 = -782.

However, it is not possible to have a negative number of lockers, so we can conclude that there is a mistake in the given information or question.