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

$24=1.1.24=1.2.12=1.3.8=1.4.6=2.2.6=2.3.4$
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 $30$
$=2\times 2\times 2=8$