# Difficult Probability Solved QuestionAptitude Discussion

 Q. Shiwani thought of a two-digit number and divided the number by the sum of the digits of the number. He found that the remainder is 3. Devesh also thought of a two-digit number and divided the number by the sum of the digits of the number. He also found that the remainder is 3. Find the probability that the two digit number thought by Shiwani and Devesh is TRUE?
 ✖ A. 1/15 ✔ B. 1/14 ✖ C. 1/13 ✖ D. 1/12

Solution:
Option(B) is correct

Let two digit number that Shiwani thought be '$xy$', where $x$ and $y$ are single digit number.

Therefore, $10x+y=p(x+y)+3$, where $p$ is a natural number

$p=\dfrac{10x+y-3}{x+y}$

Also $y+x>3$

Possible values of $x$ and $y$ for which $p$ is a natural number are:

$(x=1,y=5),(x=2,y=3),(x=3,y={1,3,5,9}),$

$(x=4,y=7),(x=5,y={1,2,9}),(x=6,y=\text{Null})$,

$(x=7,y={3,5,8},(x=8,y=\text{Null}),(x=9,y=4)$

There are 14 such two digit numbers that give a remainder of 3 when divided by the sum of the digits.

Probability that Devesh thought of the same number as Shiwani = 1/14