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Reciprocal Equation #8 (Posted on 2023-06-11)

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.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Analytical Spoiler

| Comment 1 of 4

Assume x <= y <= z

(3/2)*(3/2)*(3/2) = 27/8 which is less than 5, so x cannot be as large as 2

Thus, x must = 1, and (1+1/y)*(1+1/z) = 2.5

(3/2)*(3/2) = 9/4 which is less than 2.5, so y cannot be as large as 2

Thus, y must = 1, and (1+1/z) = 5/4 so z = 4

The only solutions to the original problem are (1,1,4), (1,4,1) and (4,1,1)

Posted by Steve Herman on 2023-06-11 07:52:16

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