Integer solutions to fraction equationI 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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Integer solutions to fraction equationI 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.

Madilyn Dunn · Accepted Answer

Let   a  ≥  bHence,  1  =      1    a    +      1    b    +      2    c    ≤      2    b    +      2    c    ,which gives  b  c  ≤  2  b  +  2  cor  (  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  aor  a  c  −  2  c  −  4  a  +  8  =  8or  (  a  −  2  )  (  c  −  4  )  =  8and 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  ) and4.  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&#039;s enough to add triples   (  b  ,  a  ,  c  )

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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