Question

If the number of integral solutions $\left(x , y , z\right)$ of the equation $xyz=18$ is $t,$ then the value of $\frac{t}{8}$ is

Answer 1

$18=1.1.18=1.2.9=1.3.6=2.3.3$ Each of $\left(1,2 , 9\right),\left(1,3 , 6\right)$ can be permuted in $3!$ ways
Each of $\left(1,1 , 18\right),\left(2,3 , 3\right)$ can be permuted in $3$ ways
So, there are $2\times 3!+2\times 3=18$ triplets $\left(x , y , z\right)$
Also, the signs are
$\left(+ , - , -\right),\left(- , + , -\right),\left(- , - , +\right),\left(+ , + , + ,\right)$
So, the desired number is $18\times 4=72$
Hence,
$\frac{72}{8}=9$