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...)?
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.
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Posted by xdog
on 2020-03-09 13:03:49 |
proof it's square
