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Period of Rational Number

Source: 2014 AMC 12A #23

February 5, 2014
modular arithmeticinductionfunctionnumber theorydivisibility testsAMC

Problem Statement

The fraction 1992=0.bn−1bn−2…b2b1b0‾,\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0}, where nn is the length of the period of the repeating decimal expansion. What is the sum b0+b1+⋯+bn−1b_0+b_1+\cdots+b_{n-1}?
<spanclass=′latex−bold′>(A)</span>874<spanclass=′latex−bold′>(B)</span>883<spanclass=′latex−bold′>(C)</span>887<spanclass=′latex−bold′>(D)</span>891<spanclass=′latex−bold′>(E)</span>892<span class='latex-bold'>(A) </span>874\qquad <span class='latex-bold'>(B) </span>883\qquad <span class='latex-bold'>(C) </span>887\qquad <span class='latex-bold'>(D) </span>891\qquad <span class='latex-bold'>(E) </span>892\qquad
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