Question

Seven digit number made up of all distinct digits 8,7,6,4,2, $x,y$ is divisible by 3. The possible number of ordered pairs $(x,y)$ is:
$
  {\text{A}}{\text{. 4}} \\
  {\text{B}}{\text{. 8}} \\
  {\text{C}}{\text{. 2}} \\
  {\text{D}}{\text{. None of these}} \\
 $

Answer 1

Hint:- In this question first we need to use the property of divisibility of a number by three. And then, put distinct values of either $x{\text{ , y}}$ other than $8,7,6,4,2$ to get the value of x and y.
Complete step-by-step answer:
We know that a number is divisible by 3 if the sum of its digits is divisible by 3.
Hence, we must have
$ \Rightarrow (8 + 7 + 6 + 4 + 2 + (x + y) = 3k$
Where k is a natural number.
$ \Rightarrow 27 + (x + y) = 3k$ ---(1)
In eq.1 27 is a multiple of 3. Now, to divide this equation by 3, $(x + y)$ should be multiple of 3.
Again since all the seven digit numbers are distinct digits therefore $x$ and $y$ must be chosen from $0,1,3,5,9$ excluding $8,7,6,4,2$ such that $(x + y)$ is a multiple of 3.
$
  {\text{If }}x{\text{ = 0 then y = 3,9 }} \\
  \therefore (0,3),(0,9){\text{ }} \\
$
$
  {\text{If }}x{\text{ = 1 then y = 5 }} \\
  \therefore (1,5){\text{ }} \\
$
$
  {\text{If }}x{\text{ = 3 then y = 0,9 }} \\
  \therefore (3,0),(3,9){\text{ }} \\
 $
$
  {\text{If }}x{\text{ = 5 then y = 1 }} \\
  \therefore (5,1){\text{ }} \\
 $
$
  {\text{If }}x{\text{ = 9 then y = 0,3}} \\
  \therefore (9,0),(9,3){\text{ }} \\
 $
Therefore, in all 8 pairs $(x,y)$ are listed above.
Hence, option B. is correct.
Note:- Whenever you get this type of question the key concept to solve this is to use the property of number divisible by given number like in this case we require the property of a number divisible by 3 that sum of its digits is divisible by three.