2013 AMC 10A Problems/Problem 13
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Problem
How many three-digit numbers are not divisible by , have digits that sum to less than , and have the first digit equal to the third digit?
Solution
These three digit numbers are of the form . We see that and , as does not yield a three-digit integer and yields a number divisible by 5.
The second condition is that the sum . When is , , , or , y can be any digit from to , as . This yields numbers.
When , we see that so . This yields more numbers.
When , so . This yields more numbers.
When , so . This yields more numbers.
When , so . This yields more numbers.
Summing, we get , .