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 [2023]

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Solution

Q.
The number of ways of selecting two numbers $a$ and $b, a \in {2, 4, 6 . . ., 100}$ and $b \in {1, 3, 5 . . ., 99}$ such that 2 is the remainder when $a + b$ is divided by 23 is [2023]

1 54

2 108

3 268

4 186

Ans.
(2)

$a \in {2, 4, 6, \dots, 100}, b \in {1, 3, 5, \dots, 99}$

$a + b = 23 λ + 2$

$λ = 0, 1, 2, \dots, 22 but λ can't be even.$

$If λ = 1, (a, b) \to 12 pairs;$ $λ = 3, (a, b) \to 35 pairs;$

$λ = 5, (a, b) \to 42 pairs;$ $λ = 7, (a, b) \to 19 pairs;$

$λ = 9, (a, b) \to 0 pairs.$

$∴$ $Total number of ways = 12 + 35 + 42 + 19 = 108$

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