If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r2t4s2, then t?

Question

If r, s, t are prime numbers and p, q are positive integers and LCM of p, q is r2t4s2, then the number of ordered pair (p, q) is ?

Answer 1

You're counting the number of ordered pairs (p,q) such that LCM(p,q) = r^2 * t^4 * s^2?

We need one more additional assumption before we start, although it seems you mean it: r, s, and t are all distinct.

Consider the exponent of r in p:

If it is 0 or 1, then the corresponding exponent in q is 2. (two cases)

If it is 2, then the corresponding exponent in q is 0,1 or 2. (three cases).

Total of five cases for the exponents of r such that we have the given LCM.

2*5-1 = 9 cases for the exponents of t

2*3-1 = 5 cases for the exponents of s.

Multiply them together to get 5*9*5 = 225 such ordered pairs (p,q).