# if r,s,t are primes . l.c.m of p,q is{( r^2)(s^4)(t^2)} . then number of possible pairs of (p,q) are -----------

mycroft holmes
272 Points
8 years ago
So p is of the form $r^{x_1}s^{y_1}t^{z_1}$ and q of the form $r^{x_2}s^{y_2}t^{z_2}$

Now, if lcm(p,q) = $r^{2}s^{4}t^{2}$ we need

$\max(x_1,x_2)=2; \max(y_1,y_2)=4; \max(z_1,z_2)=2$

For $\max(x_1,x_2)=2$ we have $x_1 \ge x_2$ or vice versa and thus 5 cases (noting that (2,2) gets counted twice).

Similarly for exponents of s, we have 5+5-1 =9 cases, and for exponents of t we have 5 cases.

Hence number of possibilities for (p,q) = 5 X 9 X 5 = 225.