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

Answer: (C)

$198$

It is given that $a_1$ and $a_2$ lies between $0$ and $9$, so let us take two random numbers say $3$ and $7$ and substitute in $D$ as follows:

$D=0.a_1{a}_2{a}_1{a}_2{a}_1{a}_2....\Rightarrow D=\mathrm{0.373737...}$

Now, since two numbers are recurring, we will multiply with $100$:

$100D=\mathrm{37.373737....}$

Now, subtract $100D$ and $D$ to get rid of recurring numbers as shown below:

$100D-D=\left(\mathrm{37.373737....}\right)-\left(\mathrm{0.373737...}\right)\Rightarrow 99D=37\Rightarrow D=\frac{37}{99}$

Therefore, the only task left is to find a multiple of $99$ in the given options, that is $198$ because $198=2\times 99$.

Hence, $198$ is the number which necessarily produces an integer when multiplied by $D$.