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# If $x$ denotes the digit at hundreds place of the number $\overline{67x19}$ such that the number is divisible by 11. Find all possible values of $x$.

Given:

$x$ denotes the digit at hundreds place of the number $\overline{67x19}$ such that the number is divisible by 11.

To do:

We have to find all the possible values of $x$.

Solution:

The number $\overline{67x19}$ is divisible by 11.

This implies,

The difference of the sums of its alternate digits will be 0 or divisible by 11.

Therefore,

Difference of $(9 + x + 6)$ and $(1 + 7)$ is zero or divisible by 11.

$15+x-8 = 0$ or multiple of 11.

$7 + x = 0$

$x = -7$ which is not possible.

If $7 + x = 11$, then $x = 11-7=4$.

The possible value of $x$ is $4$.

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