JEE Advanced · 2006 · 09 Apr 2006 · Permutations and Combinations

For the LCM of $p$ and $q$ to be $r^2{t}^4{s}^2$, each prime power in the LCM is the maximum of the corresponding powers in $p$ and $q$.

For prime $r$ with power 2: each of $p,q$ can have power 0, 1, or 2, with at least one having power 2. This gives $2\times 3-1=5$ ordered pairs.

For prime $t$ with power 4: each can have power 0, 1, 2, 3, or 4, with at least one having power 4. This gives $2\times 5-1=9$ ordered pairs.

For prime $s$ with power 2: this gives $2\times 3-1=5$ ordered pairs.

Total ordered pairs: $5\times 9\times 5=225$.