Consider three positive integers x< y< z in
Harmonic Sequence.
Determine all possible values of the positive integer constant S for which the equation 15x + Sy = 15z admits of valid solutions.
Let the three harmonic integers be
x/(a+b) < x/a < x/(ab) with x>b [a and b must both be factors of x] ***sorry I used x differently than the original problem***
So we want
15x/(a+b) + Sx/a = 15x(ab)
[15xa(ab) + Sx(a+b)(ab)  15xa(a+b)] / [a(a+b)(ab)] = 0
30xab + Sx(a^2  b^2) = 0
S = 30ab/(a^2  b^2)
So any S that fits this pattern should work, although not every choice for a and b gives an integer for S. I'm not sure what S fits this.
The simplest a=2, b=1 gives S=20
Any x (my x) that gives integers works. In this case x must be a multiple of 6.

Posted by Jer
on 20070329 11:24:46 