# Find all positive integers a, b and c such that 1/a + 1/b + 1/c = 1. Is there a finite set of solutions.

Asked on by lxsptter

justaguide | College Teacher | (Level 2) Distinguished Educator

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All the sets of positive integers (a, b, c) have to be determined such that `1/a + 1/b + 1/c = 1` .

To solve this problem start with the general assumption `a<=b<=c` .

The minimum value that a can take on is 2; if a = 1, `1/b + 1/c` would have to be equal to zero which is not possible.

For a = 2, `1/b + 1/c = 1/2` . This is possible only when b has a value greater than 2 else 1/c would have to be equal to 0. For b = 3, c = 6 and for b = 4, c = 4. If b > 4, c is less than than b.

For a = 3; b = 3 and c = 3

a cannot take on any larger values as that would make it greater than b. There is a finite set of values of the variables a, b and c that satisfies the given equation.

The solution set (a, b, c) is (2, 3, 6), (2, 4, 4) and (3, 3, 3).

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