Question

Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisible by $ 11 $ : (a) $ 92\_\_\_\_\_389 $ (b) $ 8\_\_\_\_\_9484 $ .

Answer 1

Hint: In order to find the missing number, first we should know the divisibility rule of $ 11 $ , and according to divisibility rule of $ 11 $ , if counting from left to right, the difference of the sum of the digits at the even place from the sum of the digits on the even place should be zero or a multiple of $ 11 $ .

Complete step-by-step answer:
 (a) $ 92\_\_\_\_\_389 $
Considering the missing digit to be $ a $ .
Starting from the left, the digits at the odd places are $ 9,a,8 $ and the digits at the even places are $ 2,3,9 $ .
Adding all the digits at odd places and we get: $ 9 + a + 8 = 17 + a $
Adding all the digits at even places and we get: $ 2 + 3 + 9 = 14 $ .
From the divisibility rule of $ 11 $ , we know that the difference of the sum of the odd digits and the even digits must be zero or a multiple of $ 11 $ .
So, subtracting sum of digits at even places from the sum of digits at odd places, and we get:
 $ 17 + a - 14 = 3 + a $
The difference should be zero to be a multiple of $ 11 $ , so comparing $ 3 + a $ with zero and we get:
 $ 3 + a = 0 $
Subtracting both the sides of the equation from $ 3 $ :
 $
  3 + a - 3 = 0 - 3 \\
  a = - 3 \;
  $
We obtained the value of $ a $ , but the inside digit of a number can never be negative, so the difference is not zero.
Now, the difference can be multiple of $ 11 $ , so comparing the value $ 3 + a $ with the smallest multiple of $ 11 $ , that is $ 11 $ .
 $ 3 + a = 11 $
Subtracting both the sides of the equation from $ 3 $ :
 $
  3 + a - 3 = 11 - 3 \\
  a = 8 \;
  $
Which is a positive number.
Therefore, the missing digit is $ a = 8 $ and the number becomes $ 928389 $ .
So, the correct answer is “ $ 928389 $ ”.

(b) $ 8\_\_\_\_\_9484 $
Similarly, we will solve for this number also. Considering the missing digit to be $ x $ .
Starting from left, the digits at the odd places are $ 8,9,8 $ and the digits at the even places are $ a,4,4 $ .
Adding all the digits at odd places and we get: $ 8 + 9 + 8 = 25 $
Adding all the digits at even places and we get: $ a + 4 + 4 = a + 8 $ .
From the divisibility rule of $ 11 $ , we know that the difference of the sum of the odd digits and the even digits must be zero or a multiple of $ 11 $ .
So, subtracting sum of digits at even places from the sum of digits at odd places, and we get:
 $ 25 - \left( {a + 8} \right) = 25 - a - 8 = 17 - a $
The difference should be zero to be a multiple of $ 11 $ , so comparing $ 17 - a $ with zero and we get:
 $ 17 - a = 0 $
Adding $ a $ to both the sides of the equation:
 $
  17 - a + a = 0 + a \\
 \Rightarrow a = 17 \;
 $
We obtained the value of $ a $ , but the value is a two-digit number, which can never be the missing digit.
Now, the difference can be multiple of $ 11 $ , so comparing the value $ 17 - a $ with the smallest multiple of $ 11 $ , that is $ 11 $ .
 $ 17 - a = 11 $
Adding $ a $ to both the sides of the equation:
 $
  17 - a + a = 11 + a \\
  17 = 11 + a \\
 \Rightarrow a = 17 - 11 = 6 \;
  $
Which is a positive number.
 Therefore, the missing digit is $ a = 6 $ and the number becomes $ 869484 $ .
So, the correct answer is “ $ 869484 $ ”.

Note: We can check that the value entered is correct or not by simply adding the digits at even place and subtracting it from the sum of digits at odd place, and if the obtained value is a multiple of $ 11 $ , then the number is completely divisible by $ 11 $ .
It’s important to remember the divisibility test rule of $ 11 $ .