Competition problem — Number theory

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Number theory Difficulty 4.4 AIME Prove it Estonia

Nonzero integers $a$ , $b$ and $c$ satisfy $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$ . Prove that among $a$ , $b$ , $c$ there are two integers which have a common divisor larger than 1.

Solution

Multiplying the given equation by $abc$ we get $bc + ca + ab = 0$ .

If $a$ , $b$ , $c$ were all odd, then $bc$ , $ca$ and $ab$ were also odd and their sum could not be $0$ .

If one of the numbers $a$ , $b$ , $c$ was even and the others were odd, then two of the numbers $bc$ , $ca$ and $ab$ were even and one odd, which also would not add up to $0$ .

Hence at least two of the numbers $a$ , $b$ , $c$ are even, which satisfy the conditions.

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