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 LCM 540 (Posted on 2014-12-22)
How many pairs of positive integers, without regard to order, have a least common multiple of 540?

 See The Solution Submitted by Charlie No Rating

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 One for column A, and one .. (spoiler?) | Comment 4 of 6 |
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.

 Posted by Steve Herman on 2014-12-22 16:50:37

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