If x is a digit of the number

It is given that $\overline{66784 \mathrm{x}}$ is a multiple of 9 .

Therefore, $(6+6+7+8+4+\mathrm{x})$ is a multiple of 9 .

And,

$(31+x)$ is a multiple of 9 .

Possible values of $(31+\mathrm{x})$ are $0,9,18,27,36,45, \ldots$

But $\mathrm{x}$ is a digit. So, $\mathrm{x}$ can only take value $0,1,2,3,4, \ldots 9$.

$\therefore 31+\mathrm{x}=36$

$\Rightarrow \mathrm{x}=36-31$

$\Rightarrow \mathrm{x}=5$