Question

Let $D$ be a recurring decimal of the form $D = 0.a_1 a_2 a_1 a_2 a_1 a_2 \dots$, 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 $D$?

Accepted Answer

C. We can represent $D$ as $D = \frac{10a_1 + a_2}{99}$. Thus, multiplying $D$ by any multiple of $99$ will result in an integer. Among the choices, $198 = 99 	imes 2$ is a multiple of $99$.

Recurring Decimal Integer Multiplier

Explanation

Practise this under exam conditions