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.
One for column A, and one .. (spoiler?)
