Question

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

• A. 36
• B. 24
• C. 45
• D. 30

Answer 1

Answer: (D)

$xyz =2^{3} \times 3^{1}$
Let $x=2^{\alpha_{1}} \times 3^{\beta_{1}}$
$y=2^{\alpha_{2}} \times 3^{\beta_{2}}$
$z =2^{\alpha_{3}} \times 3^{\beta_{2}}$
Now $\alpha_{1}+\alpha_{2}+\alpha_{3}=3$.
No. of non-negative intergal sol $={ }^{5} C _{2}=10$
$\& \beta_{1}+\beta_{2}+\beta_{3}=1$
No. of non-negative intergal $sol ^{ n }={ }^{3} C _{2}=3$
Total ways $=10 \times 3=30$