Write a digit in the blank space of $8__9484$ , so that the number is divisible by $11$ .

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Solution

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 $89484$
Sum of the digits at odd places = $4 + 4 += 8 +$
Sum of the digits at even places = $8 + 9 + 8 = 25$
Therefore, the difference either should be equal to $0$ or should be divisible by $11$
$\begin{array}{rcl} 25 - (8 +) & = & 0 \\ 25 - 8 - & = & 0 \\ 17 - & = & 0 \\ - () & = & - 17 (Transposing 17) \\ & = & 17 \end{array}$ or $\begin{array}{rcl} 25 - (8 +) & = & 11 \\ 25 - 8 - & = & 11 \\ 17 - & = & 11 \\ - () & = & 11 - 17 (Transposing 17) \\ - () & = & - 6 \\ & = & 6 \end{array}$
Since, $17$ is not a digit,
Hence, the required digit is $6$

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