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
• B. 108
• C. 198
• D. 288

Answer 1

The correct option is C 198

$D=0.a1a2a1a\mathrm{2....}=0.a1a2\ast (1+0.01+{0.01}^2+....+{0.01}^n+.....)$
$D=0.a1a2\ast \left(1/\right(1-0.01\left)\right)=0.a1a2/0.99=\left(a1a2\right)/99$
so, integer should be $k\ast 99$. therefore, $198=2\ast 99$