JEE MAIN - Mathematics (2023 - 30th January Evening Shift - No. 3)
The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots ., 100\}$ and $b \in\{1,3,5, \ldots . ., 99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is :
186
54
108
268
Explanation
$$a+b=23\lambda+2$$
$$\lambda=0,1,2,$$ ...., but $$\lambda$$ cannot be even as $$a+b$$ is odd
$$\lambda=1$$ $$(a, b)\to12$$ pairs
$$\lambda=3$$ $$(a,b)\to35$$ pairs
$$\lambda=5$$ $$(a,b)\to42$$ pairs
$$\lambda=7$$ $$(a,b)\to19$$ pairs
$$\lambda=9$$ $$(a,b)\to0$$ pairs
$$ \vdots $$
Total $$=12+35+42+19=108$$
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