Mathematics Question on permutations and combinations

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

Answer 1

Solution

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