The number of ways of selecting two numbers a and b , a ∈ 2 , 4 , 6 , … … , 100 and b ∈ 1 , 3 , 5 , … … , 99 such that 2 is the remainder when a + b is divided by 23 is

Step-by-step solution

Let a = 2 , 4 , 6 , . . . , 100 and b = 1 , 3 , 5 , . . . , 99 Let a + b = 23 λ + 2 If λ = 1 , then a + b = 25 So, 1 , 24 , 3 , 22 , . . . . , 23 , 2 Total ordered pairs = 12 If λ = 2 , then a + b = 48 No ordered pair possible. If λ = 3 , then a + b = 71 So, 1 , 70 ,   3 , 68 ,   5 , 66 , . . . . , 61 , 10 ,   63 , 8 ,   65 , 6 ,   67 , 4 ,   69 , 2 Total 35 ordered pairs. If λ = 5 , then a + b = 117 17 , 100 ,   19 , 98 ,   . . . ,   99 , 18 Tot