Problem:
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than ?
Answer Choices:
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E.
Solution:
One can convert to base seven by repeatedly dividing by ; the successive remainders are the digits in the baseseven representation, from right to left. Thus . It follows that the base-seven representations of positive integers less than have at most four digits, each digit is at most , and the leftmost digit is at most . If the leftmost digit is , then the remaining digits can all be for a sum of . If the leftmost digit is , then the remaining digits cannot all be . Therefore the required sum of digits cannot exceed . Because (and , the requested maximum sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions