Question

Write a digit in the blank space of $92\_\_389$, so that the number is divisible by $11$.

Answer 1

Finding the digit:

Divisibility of $11$: A number is divisible by $11$, if the difference between the sum of the digits at even places and the sum of the digits at odd places(starting from the ones place) is $0$, or divisible by $11$.

Consider the number $92\_\_389$

Sum of the digits at odd places = $9+3+2=14$

Sum of the digits at even places = $8+\_\_+9=17+\_\_$

Therefore, the difference either should be equal to $0$ or should be divisible by $11$

$\begin{array}{rcl}17+\_\_-14& =& 0\\ \_\_+3& =& 0\\ \_\_& =& -3(\mathrm{Transposing}3)\end{array}$ or $\begin{array}{rcl}17+\_\_-14& =& 11\\ \_\_+3& =& 11\\ \_\_& =& 11-3(\mathrm{Transposing}3)\\ \_\_& =& 8\end{array}$

since, $-3$ is not a digit,

Hence, the required digit is $8$