Question:
If $x$ denotes the digit at hundreds place of the number $\overline{67 x 19}$ such that the number is divisible by $11 .$ Find all possible values of $x$.
Solution:
A number is divisible by 11 , if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11 .
Sum of digits at odd places - Sum of digits at even places
$=(6+\mathrm{x}+9)-(7+1)$
$=(15+\mathrm{x})-8=\mathrm{x}+7$
$\therefore \mathrm{x}+7=11$
$\Rightarrow \mathrm{x}=4$
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