Question

The number of ways to select 2 numbers from 0,1,2,3,4 such that the sum of the square of the selected number is divisible by 5 are (repetition of digits is allowed)
$
  {\text{A}}{\text{. 9}} \\
  {\text{B}}{\text{. 11}} \\
  {\text{C}}{\text{. 17}} \\
  {\text{D}}{\text{. none of these}} \\
 $

Answer 1

Hint: We have to select 2 numbers from 0,1,2,3,4 so we have to take cases and check that in which case the sum of squares of the selected number is divisible by 5. Repetition of digits is allowed so we have to concentrate on this.

Complete step-by-step answer:
Select two numbers from $\left\{ {0,1,2,3,4} \right\} \to \left( {a,b} \right)$
Such that ${a^2} + {b^2}$ is divisible by 5.
$
  a = 0, b = 0 \Rightarrow {a^2} + {b^2} = 0 \\
  a = 1, b = 2 \Rightarrow {a^2} + {b^2} = 5 \\
  a = 2, b = 1 \Rightarrow {a^2} + {b^2} = 5 \\
  a = 3, b = 1 \Rightarrow {a^2} + {b^2} = 10 \\
  a = 1, b = 3 \Rightarrow {a^2} + {b^2} = 10 \\
  a = 2, b = 4 \Rightarrow {a^2} + {b^2} = 20 \\
  a = 4, b = 2 \Rightarrow {a^2} + {b^2} = 20 \\
    \\
 $

Hence the total number of ways are 7.
So option D is the correct option.

Note: Whenever we get this type of question the key concept of solving is we have to think step wise that means first take 0 for a and b both and check whether it is divisible or not and same with 1,2,3,4 and check whether sum of square is divisible or not if divisible then select that case and count. By doing this we can count every case.