Question

The number of triplets $\left(a,b,c\right)$ of positive integers satisfying the equation $\left|\begin{array}{ccc}{a}^3+1& a^{2}b& a^{2}c\\ a{b}^2& b^3+1& b^{2}c\\ a{c}^2& b{c}^2& c^3+1\end{array}\right|=30$ is equal to

• A. $3$
• B. $6$
• C. $9$
• D. $12$

Answer 1

Answer: (A)

$\frac{abc}{abc}\left|\begin{array}{ccc}{a}^3+1& a^{2}b& a^{2}c\\ a{b}^2& b^3+1& b^{2}c\\ a{c}^2& b{c}^2& c^3+1\end{array}\right|=30$
$\Rightarrow abc\left|\begin{array}{ccc}{a}^2+\frac{1}{a}& a^2& a^2\\ b^2& b^2+\frac{1}{b}& b^2\\ c^2& c^2& c^2+\frac{1}{c}\end{array}\right|=30$
$\Rightarrow \left|\begin{array}{ccc}{a}^3+1& a^3& a^3\\ b^3& b^3+1& b^3\\ c^3& c^3& c^3+1\end{array}\right|=30$
Apply $R_1\leftrightarrow R_1+R_2+R_3$
$\Rightarrow \left(a^3+b^3+c^3+1\right)\left|\begin{array}{ccc}1& 1& 1\\ b^3& b^3+1& b^3\\ c^3& c^3& c^3+1\end{array}\right|=30$
Apply $C_2\leftrightarrow C_2-C_1,C_3\leftrightarrow C_3-C_1$
$\Rightarrow \left(a^3+b^3+c^3+1\right)\left|\begin{array}{ccc}1& 0& 0\\ b^3& 1& 0\\ c^3& 0& 1\end{array}\right|=30$
$\Rightarrow a^3+b^3+c^3+1=30\Rightarrow a^3+b^3+c^3=29$
$\left(a,b,c\right)\equiv \left(1,1,3\right)$ or $\left(1,3,1\right)$ or $\left(3,1,1\right)$