Gọi 3 số cần tìm là \(a,b,c\left(a,b,c\in Z\right)\)
Theo đề bài ta có:
\(\dfrac{a}{b}=\dfrac{2}{3}\Rightarrow b=a:\dfrac{2}{3}=\dfrac{3}{2}a\)
\(\dfrac{a}{c}=\dfrac{4}{9}\Rightarrow c=a:\dfrac{4}{9}=\dfrac{9}{4}a\)
\(a^3+b^3+c^3=-1009\\ \Leftrightarrow a^3+\left(\dfrac{3}{2}a\right)^3+\left(\dfrac{9}{4}a\right)^3=-1009\\ \Leftrightarrow a^3+\dfrac{27}{8}a^3+\dfrac{729}{64}a^3=-1009\\ a^3\cdot\left(1+\dfrac{27}{8}+\dfrac{729}{64}\right)=-1009\\ a^3\cdot\dfrac{1009}{64}=-1009\\ a^3=\left(-1009\right):\dfrac{1009}{64}\\ a^3=\left(-1009\right)\cdot\dfrac{64}{1009}\\ a^3=-64\\ \Leftrightarrow a=-4\\ \Rightarrow b=\dfrac{3}{2}a=\dfrac{3}{2}\cdot\left(-4\right)=-6\\ \Rightarrow c=\dfrac{9}{4}a=\dfrac{9}{4}\cdot\left(-4\right)=-9\)
Vậy ba số cần tìm là -4; -6; -9
