If x denotes the digit at hundreds place of the number

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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