2013 AMC 12A Problems/Problem 10

Problem

Let $S$ be the set of positive integers $n$ for which $\tfrac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. What is the sum of the elements of $S$ ?

$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 155\qquad$

Solution

Note that $\frac{1}{11} = 0.\overline{09}$ .

Dividing by 3 gives $\frac{1}{33} = 0.\overline{03}$ , and dividing by 9 gives $\frac{1}{99} = 0.\overline{01}$ .

$S = \{11, 33, 99\}$

$11 + 33 + 99 = 143$

The answer must be at least $143$ , but cannot be $155$ since no $n \le 12$ other than $11$ satisfies the conditions, so the answer is $143$ .

2013 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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2013 AMC 12A Problems/Problem 10

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