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. Then which of the following numbers necessarily produces an integer, when multiplied by D?

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198 — D = 0.a 1 a 2 a 1 a 2 a 1 a 2 ..... &there4;100D = a 1 a 2 âa 1 a 2 ..... 100D &ndash; D = 99D = a 1 a 2 , which is an integer. &there4; If D is multiplied by 99 or any multiple of 99 it will result in an integer. From the options given, it is clear that 198 (= 99 &times; 2) will necessarily produce an integer, when multiplied by D. Hence, option (c).

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