CAT Quant Practice Problems

Question: Let D be a recurring decimal of the form D = 0. abababab ..., where digits a and b 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?
  1. 18
  2. 108
  3. 198
  4. 288

Correct Option:3

$\mathrm { D } = 0 . \overline { \mathrm { a } _ { 1 } \mathrm { a } _ { 2 } }$
Multiplied by 100 on both side
$100 \mathrm { D } = \mathrm { a } _ { 1 } \mathrm { a } _ { 2 } \cdot \mathrm { a } _ { 1 } \mathrm { a } _ { 2 }$
$100 \mathrm { D } = \mathrm { a } _ { 1 } \mathrm { a } _ { 2 } \cdot \mathrm { D }$
Therefore, $99 \mathrm { D } = \mathrm { a } _ { 1 } \mathrm { a } _ { 2 } \Rightarrow \mathrm { D } = \frac { \mathrm { a } _ { 1 } \mathrm { a } _ { 2 } } { 99 }$

Required number should be the multiple of $99 .$ So we can get an integer when multiplied by $\mathrm { D }$ .
Hence, 198 is the required number.


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