2001 AIME I Problems/Problem 1
Find the sum of all positive two-digit integers that are divisible by each of their digits.
Let our number be , . Then we have two conditions: and , or divides into and divides into . Thus or (note that if , then would not be a digit).
- For , we have for nine possibilities, giving us a sum of .
- For , we have for four possibilities (the higher ones give ), giving us a sum of .
- For , we have for one possibility (again, higher ones give ), giving us a sum of .
If we ignore the case as we have been doing so far, then the sum is .
Using casework, we can list out all of these numbers:
To further expand on solution 2, it would be tedious to test all two-digit numbers. We can reduce the amount to look at by focusing on the tens digit. First, we cannot have any number that is a multiple of . We also note that any number with the same digits is a number that satisfies this problem. This gives We start from each of these numbers and constantly add the digit of the tens number of the respective number until we get a different tens digit. For example, we look at numbers and numbers . This heavily reduces the numbers we need to check, as we can deduce that any number with a tens digit of or greater that does not have two of the same digits is not a valid number for this problem. This will give us the numbers from solution 2.
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