Question

The number of the positive integral solutions $\left(x,y,z\right)$ of the equation $xyz=24$ is $t$ , then the number of all possible factors of $t$ is

Answer 1

Answer: 8

$24=\mathrm{1.1.24}=\mathrm{1.2.12}=\mathrm{1.3.8}=\mathrm{1.4.6}=\mathrm{2.2.6}=\mathrm{2.3.4}$
Each of $\left(1,2,12\right),\left(1,3,8\right),\left(1,4,6\right),\left(2,3,4\right)$
can be permuted in $3!$ ways
Each of $\left(1,1,24\right),\left(2,2,6\right)$ can be permuted in $3$ ways
The desired number of ways is
$4\times 6+2\times 3=24+6=30$
$\Rightarrow t=30=2\times 3\times 5$
Number of factors of $30$
$=2\times 2\times 2=8$