Problem:
Given that is a 604 -digit number whose first digit is 1 , how many elements of the set have a first digit of 4 ?
Answer Choices:
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Solution:
The smallest power of 2 with a given number of digits has a first digit of 1 , and there are elements of with digits for each positive integer , so there are 603 elements of whose first digit is 1. Furthermore, if the first digit of is 1 , then the first digit of is either 2 or 3 , and the first digit of is either , or 7 . Therefore there are 603 elements of whose first digit is 2 or 3,603 elements whose first digit is , or 7 , and whose first digit is 8 or 9 . Finally, note that the first digit of is 8 or 9 if and only if the first digit of is 4 , so there are elements of whose first digit is 4 .
The problems on this page are the property of the MAA's American Mathematics Competitions