SOLUTION: 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 intege

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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