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Question Number 209116 by alcohol last updated on 02/Jul/24

Answered by A5T last updated on 02/Jul/24

$$WLOG,leta⩾b⩾c$$$$1=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}⩽\frac{3}{c}\Rightarrow c⩽3$$$$whenc=3;\frac{2}{3}=\frac{1}{a}+\frac{1}{b}⩽\frac{2}{b}\Rightarrow b⩽3$$$$b=3\Rightarrow a=3;b=2\Rightarrow a=6\to \leftarrow$$$$\Rightarrow (a,b,c)=(3,3,3)$$$$whenc=2;\frac{1}{2}=\frac{1}{a}+\frac{1}{b}⩽\frac{2}{b}\Rightarrow b⩽4$$$$b=4\Rightarrow a=4;b=3\Rightarrow a=6\Rightarrow (a,b,c)=(4,4,2);(6,3,2)$$$$whenc=1;\frac{1}{a}+\frac{1}{b}=0X$$$$\Rightarrow (a,b,c)=(3,3,3);(4,4,2);(6,3,2)$$

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