Please may I ask for your kind asistance - thank you

gavinc

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a, b and c are whole numbers. If 1/(a+(1/b+1/c))=6/23 find the value of a+b+c
 
a + (1/b + 1/c) = 3 + 5/6

Now continue......
Subhotosh, I see exactly where you are going (but will not give it away). My question to you is the solution you are leading to unique? If so, can you please prove this if you have the time?
 
Subhotosh, I see exactly where you are going (but will not give it away). My question to you is the solution you are leading to unique? If so, can you please prove this if you have the time?

Since the answers are "whole numbers" - positive integers - we can prove uniqueness through "exhaustion" (possibilities being 0, 1, 2 and 3)
 
Since the answers are "whole numbers" - positive integers - we can prove uniqueness through "exhaustion" (possibilities being 0, 1, 2 and 3)
OK, thank you. For the record I have learned quite a bit reading the posts on this forum. I hope that the help I offer is enough payment for what I am learning.
 
Subhotosh, I see exactly where you are going (but will not give it away). My question to you is the solution you are leading to unique? If so, can you please prove this if you have the time?
Not in the general case. Consider
a + (1/b + 1/c) = 3.5
 
Not in the general case. Consider
a + (1/b + 1/c) = 3.5

If by "general", you mean any real and or complex number (possible value for a, b or c) - then of course the solution may not be unique.
 
If by "general", you mean any real and or complex number (possible value for a, b or c) - then of course the solution may not be unique.
By general case I meant
Given whole numbers d and f, solutions to
a + (1/b + 1/c) = d/f
for whole numbers a, b, and c may not be unique.
 
Last edited:
a, b and c are whole numbers. If 1/(a+(1/b+1/c))=6/23 find the value of a+b+c

As a hint, lets start from Subhotosh Khan's hint but with a different problem
a + (1/b + 1/c) = 3.5 = 3 + 1/2
Since a, b, and c, are whole numbers 1/b+1/c must add to 2 or less.

(1) They can add to 2 only if both b and c are one but they must add to include the fractional part so they can't both be 1.
(2) If one of them were 1 a whole number then the other would have to be 1/2 and we have
a = 2, b = 1, c = 2 or a=2, b=2, c=1
(3) If both b and c are greater than 1, 1/b + 1/c is less than 1 and a must be 3 leaving 1/b+1/c=1/2 or 2 (b + c) = b c. So either b or c must be even. If we choose b (or c) even, b = 2 b', we have
2 b' + c =2 b' c
and we see that c (b) is also even, c = 2 c'. Thus
b' + c' = b' c'
or
c' = b' / (b'-1)
The only b' which will make c' a whole number is b'=2 which implies
a=3, b=4, c=4
 
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