Question

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

Answer 1

$\because$ The number $\overline{67x19}$ is divisible by 11
$\therefore$ The difference of the sums its alternate digits will be o or divisible by 11
$\therefore$ Difference of (9+x+6) and (1+7) is zero or dividible by 11
$\Rightarrow$ 15+x-8=0, or multiple of 11,
7+x=0 $\Rightarrow x=7$, which is not possible
$\therefore$ 7+x=11,7+x=22 etc.
$\Rightarrow x=11-7=4,x=22-7$ $\Rightarrow$ x=15 which is not a digit
$\therefore$ x=4