Find number of (x;y) with x<=y such that LCM[x;y]=600

Question

Find number of (x;y) with x≤y such that LCM[x;y]=600

 

Integer answer

Answer 1

NOTE:  REVISED BASED ON MARK M's COMMENT -- THX!

 

If you are looking for "how many pairs:"

 

The LCM[x;y] is 600 = 1*2*2*2*3*5*5.  These must be grouped such that x≤y.

     x                   y

    1             2*2*2*3*5*5

    1*2           2*2*3*5*5

    1*3           2*2*2*5

    1*2*2         2*3*5*5

    1*2*2*2      3*5*5

    . . .

       [there are 2*2*2*2*2*2 = 64 such numbers.  24≤25 is the largest x.  Then, 25≥24 starts; there is no x=y in the middle.  That means that there are 32 values of (x;y) such that x≤y.   The other 32 values have x≥y   [note: they are now interchanged from before].

 

For 600 to be the LCM[x;y], the values (1), (2*2*2), (3) and (5*5) must appear in one factor but not the other.  Thus, we have:

     x                          y

     1                 2*2*2*3*5*5

    1*2*2*2            3*5*5

   1*2*2*2*3           5*5

   1*2*2*2*3*5*5      1

In how many of these is x≤y ?

   (1;600), (8;75), (24;25)          -- There are three such pairs (x;y).

Answer 2

24 and 25 have the LCM of 600.