Ta có \(\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+2.\dfrac{1}{ab}+2.\dfrac{1}{ac}+2.\dfrac{1}{bc}=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\Leftrightarrow2\left(\dfrac{1}{ab}+\dfrac{1}{ac}+\dfrac{1}{bc}\right)=0\Leftrightarrow\dfrac{1}{ab}+\dfrac{1}{ac}+\dfrac{1}{bc}=0\Leftrightarrow\dfrac{c+b+a}{abc}=0\Leftrightarrow a+b+c=0\Leftrightarrow a+b=-c\Leftrightarrow\left(a+b\right)^3=\left(-c\right)^3\Leftrightarrow a^3+b^3+3a^2b+3ab^2+c^3=0\Leftrightarrow a^3+b^3+c^3+3ab\left(a+b\right)=0\Leftrightarrow a^3+b^3+c^3-3abc=0\Leftrightarrow a^3+b^3+c^3=3abc\)
Vì \(3abc⋮3\)
Suy ra a3+b3+c3\(⋮3\)
