How many pairs of positive integers, without regard to order, have a least common multiple of 540?
I am not 100% sure that my result is correct, but I would like to present my process of thought.
Since 540 = 2^2 *3^3 *5^1 there are 24 possible factors , each representable by a triplet (abc) the numbers in the brackets representing the powers of 2,3,5.- from(000) to (231) commas omitted
Now for a factor n, represented by (abc) the matching factors needed to produce LCM=540 are all numbers of a form (ABC) such that
max (a,A=2) and max (b,B)=3 and max (c,C)=1.
Then we count the # of matching candidates for each of the 24 from the lowest to the highest and erase the few repetitions .
The 24 factors are 1,2,3,4,5,6,9,10,12,15 , 18,20,27,30,36,45,60,90,108, 135, 180, 270, 540.
The corresponding counts are 1,1,1, 3,2 1,1,2, 3, 2 , 1,6,3,2,3,2,3,3,2,4, 2,2,1,1. -duplicity eliminated.
17+29+6=52 pairs (edited)
I hope the result is ok;
I just want to show how the list was compiled, for 36, say:
36==>220 to get a "match", only the bold digits (belonging to 231 )may be lower or equal in the "partner", which needs to contribute *31 so we get 031 131 231
i.e. 3 numbers,(15th in the list).
As to the correctness of my final number I prefer "waiting and comparing" over "rechecking and debugging".
After recounting and reading Steve's solution: I believe still 1 pair is missing : the answer should be 53 -I got 52 and do not have the time to find where is the miss.
Edited on December 22, 2014, 8:44 pm