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Precision and Accuracy

Let D be a re...

Question

Let D be a recurring decimal of the form $D = 0. a_{1} a_{2} a_{1} a_{2} a_{1} a_{2} . . . .,$ where digits a $_{1}$ and a $_{2}$ lie between $0$ and $9$ . Further, at most one of them is zero. Which of the following numbers necessarily produces an integer, when multiplied by D?

A

18

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B

108

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C

198

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D

288

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Solution

The correct option is C $198$
$D = 0. a_{1} a_{2} a_{1} a_{2} . . . . = 0. a_{1} a_{2} \times (1 + 0.01 + {0.01}^{2} + . . . . + {0.01}^{n} + . . . . .)$
$D = 0. a_{1} a_{2} \times (\frac{1}{(1 - 0.01)}) = \frac{0. a_{1} a_{2}}{0.99} = \frac{(a_{1} a_{2})}{99}$
so, integer should be $k \times 99$ .
$∴ 2 \times 99 = 198$

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