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.
The most that any two factors, such as (1+1/x)*(1+1/y), could amount to is 2*2 = 4, so the remaining factor, which could be any one of the three, has to be at least 5/4, so none of x, y or z could be larger than 4. Small enough to do with a calculator, but a computer is neater.
for x=sym(1:4)
for y=sym(1:4)
for z=sym(1:4)
if (1+1/x)*(1+1/y)*(1+1/z)==5
disp([x y z])
end
end
end
end
shows only
[1, 1, 4]
[1, 4, 1]
[4, 1, 1]
meaning two of the three variables must be 1 and the third 4.
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Posted by Charlie
on 2023-06-11 09:37:04 |
solution
