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co-prime unit fractions (Posted on 2020-03-06) Difficulty: 3 of 5
There are 3 positive integers a, b, c such that 1/c=1/a+1/b. If the greatest common divisor of a, b, c is 1, then what type of number must a+b be(e.g square number, cube number, triangular number...)?

No Solution Yet Submitted by Danish Ahmed Khan    
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proof it's square Comment 7 of 7 |
From the equation, c=(ab)/(a+b).

Any prime p factoring (a+b) will also factor ab so p will factor a, or b, or both.

But if p factors a, it must also factor b, else p won't factor (a+b).

Set a=pa' and b=pb' making the equation c=p*(a'b')/(a'+b').

So either p factors c, which violates conditions of the problem, or p factors (a'+b'), which means p^2 factors (a+b).

Thus every prime occurs an even number of times in (a+b), making it a square.

  Posted by xdog on 2020-03-09 13:03:49
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