2013 AMC 10B Problems/Problem 18
The number has the property that its units digit is the sum of its other digits, that is . How many integers less than but greater than have this property?
We take cases on the thousands digit, which must be either or : If the number is of the form where are digits, then we must have Since we must have By casework on the value of , we find that there are possible pairs , and each pair uniquely determines the value of , so we get numbers with the given property.
If the number is of the form then it must be one of the numbers Checking all these numbers, we find that only has the given property. Therefore, the number of numbers with the property is
This solution picks up from finding that in solution 1.1. Instead of using casework to find all possible pairs, , let's introduce a dummy variable, . Let us now have that , where are all nonnegative.
We may now use stars and bars to distribute units between and . Any units that is given will essentially be discarded - this is how we get the 'less than' in the 'less than or equal to ' relation we found earlier.
Using two dividers, we find that the number of distributions is which is . We proceed from here as above.
Solution 2 (Casework)
Let's start with the case that starts with . We have only one number, which is . If we look at the , we have no solutions because , and because we can only use digits from through , it is impossible. If we looks at the , we do have one solution, which is . If we look a the , we have solutions, namely, and .
We can see a pattern here. The pattern is every hundred you go down, you have more solution. Therefore, we have which is =
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