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) ??
Note that . Therefore one of a or b must be divisible by , and one of a or b must be divisible by . How many ordered pairs (a,b) satisfy these conditions?
This method definitely generalizes, but I doubt there is a general formula since it becomes one giant counting problem when N has more prime divisors.
We can. If the maximum power of 2 that divides either a or b is, say, 3, then there is no reason to include 2^4 as a factor of LCM(a, b). In the notation of post #4, since 2^4 is the maximum power of 2 that divides 144, we have max(x, y) = 4, so x = 4 or y = 4 or both. This means that 2^4 divides a or b or both. Similarly for 3^2.