Rupa Reddy
2

Let two digit number that Shiwani thought be 'xy

xy', where x

x and y

y are single digit number.

Therefore, 10x+y=p(x+y)+3

10x+y=p(x+y)+3, where p

p is a natural number

p=(10x+y−3)/(x+y)


p=10x+y−3x+y

Also y+x>3

y+x>3

Possible values of x

x and y

y for which p

p is a natural number are:

(x=1,y=5),(x=2,y=3),(x=3,y=1,3,5,9),

(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=Null)

(x=4,y=7),(x=5,y=1,2,9),(x=6,y=Null),

(x=7,y=3,5,8,(x=8,y=Null),(x=9,y=4)

(x=7,y=3,5,8,(x=8,y=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