Có \(\left(x+y+z\right)^3-\left(x^3+y^3+z^3\right)\)
\(=\left[\left(x+y\right)+z\right]^3-\left(x^3-y^3-z^3\right)\)
\(=\left(x+y\right)^3+3\left(x+y\right)^2z+3\left(x+y\right)z^2+z^3-\left(x^3+y^3+z^3\right)\)
\(=3xy\left(x+y\right)+3\left(x+y\right)^2z+3\left(x+y\right)z^2\)
\(=3\left(x+y\right)\left[xy+\left(x+y\right)z+z^2\right]\)
\(=3\left(x+y\right)\left[x\left(y+z\right)+z\left(y+z\right)\right]\)
\(=3\left(x+y\right)\left(y+z\right)\left(x+z\right)\)
Do x,y,z nguyên và cùng tính chẵn lẻ \(\Rightarrow\left(x+y\right);\left(y+z\right);\left(z+x\right)\) đều là ba số chẵn
\(\Rightarrow\left(x+y\right)\left(y+z\right)\left(z+x\right)⋮8\)
mà (3;8)=1 và 3.8=24
\(\Rightarrow3\left(x+y\right)\left(y+z\right)\left(z+x\right)⋮24\) (đpcm)
