Question

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

• A. 54
• B. 108
• C. 268
• D. 186

Answer 1

Answer: (B)

108

Answer 2

$a∈{2,4,6,8,10,….,100}$
$b∈{1,3,5,7,9,……,99}$
Now, $a+b∈{25,71,117,163}$

(i) $a+b=25$, no. of ordered pairs $(a, b)$ is $12$
(ii) $a+b=71$, no. of ordered pairs $(a, b)$ is $35$
(iii) $a+b=117$, no. of ordered pairs $(a, b)$ is $42$
(iv) $a+b=163$, no. of ordered pairs $(a, b)$ is $19$

$∴ total =108 \;pairs$

$\text{Therefore, the correct option is (B):} 108$