Question 9

If $$k$$ is a positive integer, let $$D_k$$ denote the ultimate sum of digits of $$k$$. That is, if $$k$$ is a digit, then $$D_k = k$$. If not, take the sum of digits of $$k$$. If this sum is not a single digit, take the sum of its digits. Continue this process until you obtain a single digit number. By $$D_k$$ we mean this single digit number. $$\{D_p \mid p \text{ is a positive multiple of } 2007\}$$ = _________, in roster form. (The set has a single element; give it.)


Correct Answer: 9

Solution

The repeated digit sum of a positive integer is its remainder on division by 9, with 9 in place of 0. Since $$2007 = 9 \times 223$$, every multiple of 2007 is a positive multiple of 9, so its ultimate digit sum is 9. The set is $$\{9\}$$.

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