Question

The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is :

• A. $36$
• B. $45$
• C. $24$
• D. $30$

Answer 1

The correct option is D $30$

$x.y.z=24$
$x.y.z=2^3{.3}^1$
$x=2^{a_1}\cdot 3^{b_1},y=2^{a_2}\cdot 3^{b_2},z=2^{a_3}\cdot 3^{b_3}$
$a_1,a_2,a_3\in \{0,1,2,3\}$
$b_1,b_2,b_3\in \{0,1\}$
Case: $1$ $a_1+a_2+a_3=3$
Non negative solution $={}^{3+3-1}{C}_{3-1}{=}^5{C}_2=10$
Case: $2$ $b_1+b_2+b_3=1$
Non negative solution $={}^{1+3-1}{C}_{3-1}{=}^3{C}_2=3$
$\therefore$ Total solutions $=10\times 3=30$