Determine the total number of triplets (x, y, z) of positive integers that satisfy this system of equations:

lcm(x, y) = 1000 and:
lcm(y, z) = 2000 and:
lcm(z, x) = 2000

(In reply to computer solution by Charlie)

I've counted the possible combinations manually and got 68.<o:p></o:p>

Prior to publishing my solution  I saw the computerized solution and after a while got my list corrected.<o:p></o:p>

My notation was as follows:<o:p></o:p>

f- any factor of 1000 i.e. 1,2,4,...1000<o:p></o:p>

K- 1000; <o:p></o:p>

 2K- 2000; <o:p></o:p>

S- each of 16,80,400; <o:p></o:p>

R- each of 8,40,200; <o:p></o:p>

Z-each of 125,250,500; <o:p></o:p>

The x,y,z triplets:<o:p></o:p>

f,K,2K..................16*2=32<o:p></o:p>

K,K,2K........................-1<o:p></o:p>

K,K,S........................3*1=3<o:p></o:p>

 R,Z,2K...................9*2=18<o:p></o:p>

Z,K,S...................9*2=18<o:p></o:p>

 <o:p></o:p>

The total (symmetry included) 34+36=70<o:p></o:p>

The total (symmetry excluded) 16+3+9+9=37<o:p></o:p>

 <o:p></o:p>

It took me more than 2 hrs to count and to debug- hope

no errors were introduced while pasting.

Posted by Ady TZIDON on 2014-02-14 17:34:27