The landlord of an apartment building needs to purchase enough digits to label all of the apartments from 100 through 125 on the first floor and 200 through 225 on the second floor. The digits can only be purchased in a package that contains one of each digit 0 through 9. How many packages must the landlord purchase?
Since 1 and 2 are used at least once in half of the apartments, and no other number is used this often, either 1 or 2 will be the most frequent digit used.

Notice, though, that since all of the numbers of the form $\star1\star$ appear but only 6 of the numbers $\star2\star$ appear, 2 will be used less often than 1 and we should count the number of 1s to find the number of packages needed.

100 through 125 requires 26 ones just for the hundreds place. 100 through 125 and 200 through 225 require the same number of ones for the tens and units places; that is, thirteen.

So, there are $26 + 2 \cdot 13 = 52$ ones used. Therefore, the landlord must purchase $\boxed{52}$ packages.
52
Level 5
Prealgebra
Problem: The landlord of an apartment building needs to purchase enough digits to label all of the apartments from 100 through 125 on the first floor and 200 through 225 on the second floor. The digits can only be purchased in a package that contains one of each digit 0 through 9. How many packages must the landlord purchase?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the total number of packages needed, first identify the range of apartment numbers that require labeling. The apartments are numbered from 100 to 125 on the first floor, which includes 26 apartments. The second floor has apartments numbered from 200 to 225, adding another 26 apartments. Therefore, the total number of apartments is 26 + 26 = 52.

Next, analyze the digits required for labeling these apartments. The apartment numbers from 100 to 125 use the digits 1, 0, 2, 3, 4, and 5. Specifically, the digit 1 appears in every number from 100 to 125, the digit 0 appears in 100, the digit 2 appears in 200 to 225, and the digits 3, 4, and 5 appear in their respective numbers. 

For the second floor, the apartments from 200 to 225 also require the digits 2, 0, 1, 3, 4, and 5. The digit 2 appears in every number from 200 to 225, while the digits 0, 1, 3, 4, and 5 appear as needed.

Now, count the frequency of each digit across both ranges. The digits 0 through 9 must be accounted for, and since the landlord can only purchase packages containing one of each digit, the total number of packages needed will be determined by the maximum frequency of any digit used.

The digit 1 is used 26 times (from 100 to 125), the digit 2 is used 26 times (from 200 to 225), and the digits 0, 3, 4, and 5 are used less frequently. The digits 6, 7, 8, and 9 are not used at all.

Since each package contains one of each digit from 0 to 9, and the maximum frequency of any digit needed is 26 (for digits 1 and 2), the landlord must purchase enough packages to cover this maximum frequency. Therefore, the landlord needs to buy 52 packages to ensure that all apartments can be labeled correctly. 

Thus, the final answer is 52.<|END_OF_LATENT|> The final answer is $52$.