Problem:
Let denote the smallest positive integer that is divisible by both and , and whose base- representation consists of only 's and 's, with at least one of each. What are the last four digits of ?
Answer Choices:
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Solution:
Since is divisible by , the sum of the digits of must be a multiple of . At least one digit of is , so at least nine digits must be , and at least one digit must be . For to be divisible by , the last two digits of must each be . These conditions are satisfied by several ten-digit numbers, of which the smallest is .
The problems on this page are the property of the MAA's American Mathematics Competitions