Question 501906: Show that there is only one set of different positive integers, x,y,z that
1= 1/x+1/y+1/z
ie 1 can be expressed as the sum of the reciprocals of the 3 different positive integers in only one way.
Deduce that if n is any odd integer greater than 3, then i can be expressed as the sum of n reciprocals of different positive integers.
For which even integers is this possible? Justify your answer
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website! 
Without loss of generality we can assume x < y < z.
Everybody knows that
where x = y = z = 3 would be a solution
But that is not acceptable because x, y, and z must all be
different. x, y, and z cannot all be 3 or larger than
3 because the sum would then be less than 1. So x
is a denominator that is less that 3. Then it cannot be 1
because the right side  would then be > 1.
So x = 2, and we have:
Therefore
Now everybody also knows that
and y = z = 4 would be a solution
But that is also not acceptable because y and z must be
different. So y and z cannot both be larger than
4 because the sum would then be less than  .
So let y be the one that is less than 4. y cannot be 1
because that would make the left side be > . y
cannot be 2 because x=2. So y=3, Therefore
becomes
Multiply through by 6z
So
 and so {x,y,z}={2,3,6}
I'll see if I can finish the other part later.
Edwin
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