Question 127

Let D be recurring decimal of the form, $$D = 0.a_1a_2a_1a_2a_1a_2...$$, where digits $$a_1$$ and $$a_2$$ lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?


Case 1: $$a_1=0$$
So, D equals $$0.0a_20a_20a_2...$$
So, 100D equals $$a_2.0a_20a_2...$$
So, 99D equals $$a_2$$


Case 2: $$a_2=0$$
So, D equals $$0.a_10a_10a_1...$$
So, 100D equals $$a_10.a_10a_1....$$
So, 99D equals $$a_10$$

So, in both the cases, 99D is an integer. From the given options, only option C satisfies this condition (198=2*99) and hence the correct answer is C. 

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