Do there exist positive integers a , b , c such that 1 / a + 1 / b + 1 / c = 47 / 48?So far the only conclusion i have drawn is that a + b + c is greater than 48. However, I cannot make anymore connections that would help me solve the problem. Am I just supposed to "brute force" this by bashing the algebra, or is there some logic I am missing?

Question

Do there exist positive integers   a  ,  b  ,  c such that   1      /    a  +  1      /    b  +  1      /    c  =  47      /    48?So far the only conclusion i have drawn is that   a  +  b  +  c is greater than 48. However, I cannot make anymore connections that would help me solve the problem. Am I just supposed to &quot;brute force&quot; this by bashing the algebra, or is there some logic I am missing?

indivisast7 · Accepted Answer

The hint.Let   a  ≥  b  ≥  cHence,      47    48    =      1    a    +      1    b    +      1    c    ≤      3    c    ,which gives  c  ≤            3      ⋅      48        47  and from here  c  ≤  3.Also, it&#039;s obvious that   c  ≥  2Now, prove that for c=2 and for c=3 it&#039;s impossible.For example, let   c  =  3Hence,      31    48    =      1    a    +      1    b    ≤      2    b    ,which gives   b  ≤  3 and since   b  ≥  c, we obtain   b  =  3, which is impossible.

Do there exist positive integers a,b,c such that 1/a+1/b+1/c=47/48?

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