Problem:
There are exactly positive integers with such that the base- integer is divisible by 16 (where 16 is in base ten). What is the sum of the digits of ?
Answer Choices:
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Solution:
Notice that , and consider the residue classes of this number modulo 8 . If , then , and if or 6 , then . In each case is divisible by 16 .
In all other cases is not divisible by 16 . Indeed, if , or , then is odd, and , so is divisible by no power of 2 greater than . If or , then and are both odd multiples of 2 , so is divisible by 8 , but not by 16 .
Because , there are positive integers that are congruent to 3 , 6 , or 7 modulo 8 . The number 3 must be excluded from this total, because the problem statement requires to be at least 5 . Thus , and the sum of the digits of is .
The problems on this page are the property of the MAA's American Mathematics Competitions