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 $so{l}^n={}^3{C}_2=3$
Total ways $=10\times 3=30$