Question

If $r,s,t$ are prime numbers and $p,q$ are the positive integers such that LCM of $p,q$ is $r^2{s}^4{t}^2$ then the number of ordered pairs $(p,q)$ is

• A. 252
• B. 254
• C. 225
• D. 224

Answer 1

Answer: (C)

Since, $r,s,t$ are prime numbers.
$\therefore$ Selection of $p$ and $q$ are as under

p	q	Number of ways
$r^0$	$r^2$	1 way
$r^1$	$r^2$	1 way
$r^2$	$r^0,r^1,r^2$	3 ways

$\therefore$ Total number of ways to select, $r=5$
Selection of $s$ as under

$s^0$	$s^4$	1 way
$s^1$	$s^4$	1 way
$s^2$	$s^4$	1 ways
$s^3$	$s^4$	1 ways
$s^4$		15 ways

$\therefore$ Total number of ways to select $s=9$
Similarly, the number of ways to select $t=5$
$\therefore$ Total number of ways $=5\times 9\times 5=225$