Problem:
For each positive integer , let denote the sum of the digits of . For how many values of is
Answer Choices:
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E.
Solution:
If , then . If , then . Therefore if satisfies the required condition it must also satisfy
In addition, , and all leave the same remainder when divided by 9. Because 2007 is a multiple of 9 , it follows that , and must all be multiples of 3 . The required condition is satisfied by 4 multiples of 3 between 1969 and 2007, namely 1977, 1980, 1983, and 2001.
Note: There appear to be many cases to check, that is, all the multiples of 3 between 1969 and 2007. However, for , we have , so these numbers are eliminated. Thus we need only check 1971, 1974, 1977, 1980, 1983, 1986, 2001, and 2004.
The problems on this page are the property of the MAA's American Mathematics Competitions