Even i am getting 360

 

Basically, the numbers r², s⁴ and t² have to appear in either number. This can be assigned in 2*2*2 = 8 ways.

 

Now if r² appears in one number, the other can have r⁰ or r¹ or r² as its factors.

 

This reasoning gives 8*3*5*3 = 360 such ordered pairs