Find all possible triplets (x, y, z) of positive integers that satisfy this equation:
(1+1/x)*(1+1/y)*(1+1/z) = 5
Prove that these are the only possible triplets in conformity with the given conditions.
Note: Computer program solutions are welcome, but an analytic solution is preferred.
Order the unknowns so that x <= y <= z. Then
1 + 1/x >= 1 + 1/y >= 1 + 1/z. Therefore
(1 + 1/x)^3 >= 5 and x <= 1/(5^[1/3] - 1) < 1.41
Since x must be a positive integer, this gives x = 1
Next we have [(y + 1)/y] * [(z + 1)/z] = 5/2
so [(y + 1)/y]^2 > 5/2.
Therefore y <= 1/( [5/2]^.5 - 1 ) < 1.7
and since y must be a positive integer, this gives y = 1.
This leaves
2 * 2 * (z+1)/z = 5.
So that z = 4.
This is the sole solution.
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Posted by FrankM
on 2023-06-11 12:07:34 |
Analytic solution
