Question

The number of ways of selecting two numbers $a$ and $b,a\in \{2,4,6,\dots .,100\}$ and $b\in \{1,3,5,\dots ..,99\}$ 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)

$a\in \{2,4,6,8,10,\dots .,100\}$
$b\in \{1,3,5,7,9,\dots \dots ,99\}$
Now, $a+b\in \{25,71,117,163\}$
(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
$\therefore$ total $=108$ pairs