Question:
If $x$ is a digit of the number $\overline{66784 x}$ such that it is divisible by 9 , find possible values of $x$.
Solution:
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 $0,1,2,3,4, \ldots 9$.
$\therefore 31+\mathrm{x}=36$
$\Rightarrow \mathrm{x}=36-31$
$\Rightarrow \mathrm{x}=5$