Problem:
A positive integer has base-eleven representation and base-eight representation , where , and represent (not necessarily distinct) digits. Find the least such expressed in base ten.
Solution:
The conditions of the problem imply that , so . is a digit in base and , so must be or . Since we want the smallest , we try the smallest value of at . It follows that , which is . We see must be or . If , then is not an integer, but if , then , so . Thus the smallest value of , and .
The problems on this page are the property of the MAA's American Mathematics Competitions