Question

$D$ is a recurring decimal of type $0.\ a_1a_2a_1a_2a_1a_2 \dots$ Here $a_1$ and $a_2$ are single digit numbers between $0$ and $9$. This number $D$, when multiplied by which of the following numbers gives a product which is an integer?

• A. 18
• B. 108
• C. 198
• D. 288

Hint:

A recurring decimal with a block length of $n$ digits is a rational number with denominator $10^n - 1$.

Answer 1

The Correct Option is C

A recurring decimal of the form $0.\overline{a_1a_2}$ can be expressed as: \[ D = \frac{\text{two-digit number } a_1a_2}{99} \] Multiplying by $99$ makes it an integer. But here, the repeat is $a_1a_2a_1a_2$ which is length $2$. We want the smallest option divisible by $99$. Check: $198 \div 99 = 2$ (integer) $\Rightarrow$ works. $18, 108, 288$ are not multiples of $99$. \[ \boxed{198} \]