I am having trouble finding all solutions (or at least proving I have all the solutions already). The equation is 1/a+1/b+2/c=1 *a,b,c are positive I tried to base it on the solutions of a similar equation with the last term a 1 instead of a 2.

proximumha 2022-08-12 Answered
Integer solutions to fraction equation
I am having trouble finding all solutions (or at least proving I have all the solutions already).
The equation is
1 a + 1 b + 2 c = 1
*a,b,c are positive I tried to base it on the solutions of a similar equation with the last term a 1 instead of a 2.
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Answers (1)

Madilyn Dunn
Answered 2022-08-13 Author has 16 answers
Let a b
Hence,
1 = 1 a + 1 b + 2 c 2 b + 2 c ,
which gives
b c 2 b + 2 c
or
( b 2 ) ( c 2 ) 4.
Now, if b 3 we obtain:
c 2 4 b 2 4 ,
which gives c 6 and the rest is smooth.
For example, the case b = 1 is impossible, but for b = 2 we obtain
1 2 = 1 a + 2 c
or
( b 2 ) ( c 2 ) 4.
Now, if b 3 we obtain:
c 2 4 b 2 4 ,
which gives c 6 and the rest is smooth.
For example, the case b = 1 is impossible, but for b = 2 we obtain
1 2 = 1 a + 2 c
or
a c = 2 c + 4 a
or
a c 2 c 4 a + 8 = 8
or
( a 2 ) ( c 4 ) = 8
and solve some systems:
1. a 2 = 1 and c 4 = 8 gives ( 3 , 2 , 12 )
2. a 2 = 2 and c 4 = 4 gives ( 4 , 2 , 8 )
3. a 2 = 4 and c 4 = 2 gives ( 6 , 2 , 6 ) and
4. a = 2 = 8 and c 4 = 1 gives ( 10 , 2 , 5 ) .
Now, for a b 3 c { 3 , 4 , 5 , 6 }, which is for you.
For the full ending of the solution it's enough to add triples ( b , a , c )

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