Question

If $r, s, t$ are prime numbers and p , q are the positive integers such th a t 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

The Correct Option is C

Since, r, s, t are prime numbers,
$\therefore \, \, $Selection of p and q are as under
p $\hspace10mm $ q $\hspace10mm $ Number of ways
$r^0$ $\hspace10mm $ $r^2$ $\hspace10mm $ 1 way
$r^1$ $\hspace10mm $ $r^2$ $\hspace10mm $ 1 way
$r^2$ $\hspace10mm $ $r^0,r^1,r^2$ $\hspace7mm $ 3 way
$\therefore \, $ Total number of ways to select, r = 5
Selection of s as under
$s^0$ $\hspace10mm $ $s^4$ $\hspace10mm $ 1 way
$s^1$ $\hspace10mm $ $s^4$ $\hspace10mm $ 1 way
$s^2$ $\hspace10mm $ $s^4$ $\hspace10mm $ 1 way
$s^3$ $\hspace10mm $ $s^4$ $\hspace10mm $ 1 way
$s^4$ $\hspace25mm $ 5 way
$\therefore \, $ Total number of ways to select s = 9
Similarly, the number of ways to select t = 5
$\therefore \, $ Total number of ways = 5 x 9 x 5 = 225