Question

Tìm các bộ số nguyên a, b, c thỏa mãn: \(\frac{1}{a^3}\)+\(\frac{1}{b^3}\)+\(\frac{1}{c^3}\)+\(\frac{3}{4}\)=0 và ab+bc+ca=0.

Answer 1

ĐKXĐ: \(abc\ne0\)

Đặt \(\left(\frac{1}{a};\frac{1}{b};\frac{1}{c}\right)=\left(x;y;z\right)\Rightarrow x+y+z=0\Rightarrow x^3+y^3+z^3=3xyz\)

\(\Rightarrow x^3+y^3+z^3+\frac{3}{4}=0\Leftrightarrow3xyz+\frac{3}{4}=0\)

\(\Leftrightarrow xyz=-\frac{1}{4}\Leftrightarrow\frac{1}{xyz}=-4\Leftrightarrow abc=-4\)

\(\Rightarrow ab=\frac{-4}{c}\Rightarrow c=Ư\left(4\right)=\left\{-4;-2;-1;1;2;4\right\}\)

\(c=-4\Rightarrow ab=1\Rightarrow\left(a;b\right)=\left(1;1\right);\left(-1;-1\right)\)

\(c=-2\Rightarrow ab=2\Rightarrow\left(a;b\right)=\left(1;2\right);\left(2;1\right);\left(-2;-1\right);\left(-1;-2\right)\)

....

Tương tự