Number of ordered pairs with given lcm Suppose $(p,q)$ is an ordered pair of  natural numbers with lcm $r^2.s^4.t^2$, where $r,s,t$ are distinct primes.We have to find the number of all such ordered pairs.
One simple idea I tried is to fix p and then take all possible values of q s.t. the lcm is preserved.But my calculations get cumbersome.Would somebody please help me?Thank u. 

Either $p$ or $q$ must have a factor of $r^2$. The other one can have either $r^0, r^1$ or $r^2$. That's five ways to the distribute $r$-factors. Similarily, the $s$-factors may be distributed in $9$ ways, and the $t$ factors can be distributed in $5$ ways. All in all we get $5.9.5 = 225$ different pairs.
