If N= 144 How many sets of two values (a,b) are there for which the LCM (a,b) is 144?

What the general formula for this (with proof)?

I think what's required for 2 numbers to have a LCM of N is that neither of the numbers can share common prime factors. For example if the prime factors of N are a,a,b,b,b,c then possible ways to combine into two integers that don't share any prime factors are:

axa, bxbxbxc
axaxbxbxb, c
axaxc, bxbxb

Example: N = 360 = 2x2x2x3x3x5, so factors with LCM =360 are:
2x2x2 = 8 and 3x3x5 = 45
2 x 2 x2 x 5 = 40 and 3 x 3 = 9
2x2x2x3x3 = 72 and 5

Try this for N=144 and post back with what you find. As for the proof of this - please show us your attempt and we'll provide some feedback.