Problem:
How many positive integers less than 1000 are 6 times the sum of their digits?
Answer Choices:
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Solution:
The only such number is 54. A single-digit number would have to satisfy , implying , which is impossible. A two-digit number would have to satisfy , so and then necessarily and ; hence the number is 54 . A three-digit number would have to satisfy or . But the left side of the expression is at least 94 while the right side of the expression is at most 45 , so no solution is possible.
The problems on this page are the property of the MAA's American Mathematics Competitions