How many pairs of positive integers, without regard to order, have a least common multiple of 540?

Without enumerating, I come up with 105.

540 = 5 * 2^2 * 3^3

The powers of 5 must be distributed between the two numbers as follows:

1 5 or

5 5 or

5 1 3 different combinations (this is 2n + 1), where n is is exponent of the factor of 5

The powers of 2 must be distributed between the two numbers as follows:

1 4 or

2 4 or

4 4 or

4 2 or

4 1 5 different combinations (this is 2n + 1), where n is is exponent of the factor of 2

Similarly, the powers of 3 must be distributed between the 2 numbers in 7 different ways

Altogether, the number of total combinations is 3 * 5 * 7 = 105

This is bigger than previously submitted solutions, but I have not investigated who is right and who is not.