2009 AMC 12A Problems/Problem 12
Problem
How many positive integers less than are times the sum of their digits?
Solution
The sum of the digits is at most . Therefore the number is at most . Out of the numbers to the one with the largest sum of digits is , and the sum is . Hence the sum of digits will be at most .
Also, each number with this property is divisible by , therefore it is divisible by , and thus also its sum of digits is divisible by .
We only have five possibilities left for the sum of the digits: , , , , and . These lead to the integers , , , , and . But for the sum of digits is , which is not , therefore is not a solution. Similarly we can throw away , , and , and we are left with just solution: the number .
See Also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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All AMC 12 Problems and Solutions |