Find all sets of integers A, B, and C which satisfy the following:
1/A + 1/B + 1/C = 1.
(In reply to
partial solution by Ender)
This is to prove that all solutions involving negative integers are of the set (-X,1,X).
First, there can be no solutions where all three are negative, since three negatives would sum to a negative value, and we need a sum of 1. Thus, no solutions involve three negative numbers.
Second, there can be no solutions where two numbers are negative. If A and B were both negative, the largest value 1/C can have is 1 (when C = 1). That would mean 1/A + 1/B would need to sum to 0 or greater, but two negatives can only sum to a negative number. Thus, no solutions involve two negative numbers.
Third, We need to prove that there is only one solution to 1/A+1/B+1/C=1 for a negative A. We can rewrite this as 1/B+1/C=1+1/A, and all A, B,and C must now be greater than 0.
We want solutions other than (-X,1,X), so neither B or C can now 1. We can write that as 1 < B and 1 < C. This gives us: 1/B + 1/C = 1 + 1/A; 1 < B; 1 < C; 1 <= A
Thus, B and C can be no smaller than 2, so the largest value on the left hand side is 1/2 + 1/2 = 1. But we need this to be 1 + 1/A! Thus, there are no valid solutions involving negative numbers that are not of the set (-X,1,X).
So my previous answer stands as complete.
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Posted by Ender
on 2003-07-18 04:20:33 |
re: partial solution
