Problem:
Find the three-digit positive integer whose representation in base nine is , where , and are (not necessarily distinct) digits.
Solution:
Answer (227):
The given conditions imply that , so . Modulo 71 this equation becomes , so is a multiple of 71 . Because , it must be that because 0 is the only multiple of 71 between -18 and 63 . Because cannot be 0 and 7 must divide , it follows that , from which and . The requested three-digit integer is 227 .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions