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

Answer: 9

$18=\mathrm{1.1.18}=\mathrm{1.2.9}=\mathrm{1.3.6}=\mathrm{2.3.3}$ 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,
$\frac{72}{8}=9$