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jmc

number theory senior

Problem

A textbook has 1,000 pages. How many of the pages have page numbers whose digits add up to exactly 4?
Solution
There are only a few ways to make one, two, and three digit numbers add up to . The only one-digit number whose sum is is itself. Continuing with two-digit numbers, we note that the digits must be and , and , or and . This means that , , , and are the only two-digit numbers whose digits sum to 4. For the three-digit numbers, we organize the work in a table.

Adding up the last column, we see that there are three-digits numbers whose digits add up to . Adding those to the possible one-and-two digit numbers, we get pages in the textbook which have digits that add up to .
Final answer
15