Question

Total number of positive integral solutions of the equation $x_1\cdot x_2\cdot x_3-60$, is equal to

• A. 27
• B. 54
• C. 64
• D. None of these

Answer 1

Answer: (B)

Considering the equation $x_1{x}_2{x}_3=60=(2{)}^2$. (3) ${}^1$. (5)
Let $x_i=2^{\alpha i},3^{\beta i},4^{\gamma i},\alpha i,\beta i,\gamma i\ge 0$
$\Rightarrow (2{)}^{{\alpha }_i+{\alpha }_2+{\alpha }_3}\cdot (3{)}^{{\beta }_i+{\beta }_2+{\beta }_3}\cdot (5{)}^{{\gamma }_1+{\gamma }_2+{\gamma }_3}\cdot (2{)}^2\cdot (3{)}^1\cdot (5{)}^1$
$\Rightarrow {\alpha }_1+{\alpha }_2+{\alpha }_3=2;{\beta }_1+{\beta }_2+{\beta }_3=1;{\gamma }_1+{\gamma }_2+{\gamma }_3=1$
Total number of positive integral solutions =
${}^4{C}_2\cdot {}^3{C}_2\cdot {}^3{C}_2=6\times 3\times 3=54$