Determine
all possible ordered triplets (x,y,z) of
nonzero integers that satisfy this equation:
1 1 1
-- + ---- + ----- = 1
x xy xyz
[Edit: the following answer is in error in that I only considered positive values rather than nonzero values. ]
(3,1,1) and (2,2,1) are the only solutions I can find.
multiply both sides by xyz
yz + z + 1 = xyz
z + 1 = yz * (x-1)
y * (x-1) = (z+1)/z which is never an integer unless z=1
Now that we know that z = 1, the equation becomes
1/x + 2/xy = 1
y * (x-1) = 2
y and (x-1) can only be (1,2) or (2,1) each of which gives one of the above solutions.
So these are the only solutions.
Edited on July 16, 2023, 10:18 am
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Posted by Larry
on 2023-07-16 08:49:28 |
Solution
