How many pairs of positive integers, without regard to order, have a least common multiple of 540?

There are 53 pairs of integers that have an LCM of 540:
(  1, 540), (  2, 540), (  3, 540), (  4, 135),
(  4, 270), (  4, 540), (  5, 108), (  5, 540),
(  6, 540), (  9, 540), ( 10, 108), ( 10, 540),
( 12, 135), ( 12, 270), ( 12, 540), ( 15, 108),
( 15, 540), ( 18, 540), ( 20,  27), ( 20,  54),
( 20, 108), ( 20, 135), ( 20, 270), ( 20, 540),
( 27,  60), ( 27, 180), ( 27, 540), ( 30, 108),
( 30, 540), ( 36, 135), ( 36, 270), ( 36, 540),
( 45, 108), ( 45, 540), ( 54,  60), ( 54, 180),
( 54, 540), ( 60, 108), ( 60, 135), ( 60, 270),
( 60, 540), ( 90, 108), ( 90, 540), (108, 135),
(108, 180), (108, 270), (108, 540), (135, 180),
(135, 540), (180, 270), (180, 540), (270, 540),
(540, 540)

The prime factors of 540 (with the number of distinct primes indicated as exponents and the primes separated by the multiplication signs) are 5×3³×2². Each of the numbers of the number pairs are composed solely of the factors of 540. Where the number of prime factors of the smaller number that are absent in the larger, when multiplied to the larger will result in the number 540.

Edited on December 24, 2014, 11:56 am

Posted by Dej Mar on 2014-12-22 09:50:10