Write a digit in the blank space of each of the following numbers so that thenumber formed is divisible by 11.i. 12_268ii. 2755_96

To determine the digit that should be placed in the blank space so that the resulting number is divisible by 11, we can use the rule for divisibility by 11. According to this rule, a number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11.

Let's solve each part step by step:

Part i: 12_268

1. Identify the positions of the digits:

   • 1 (odd position)
   • 2 (even position)
   • _ (odd position, blank space)
   • 2 (even position)
   • 6 (odd position)
   • 8 (even position)

2. Sum of digits in odd positions (1, _, 6):

   • Let the blank space be represented by $x$.
   • Sum of odd positions: $1 + x + 6 = 7 + x$

3. Sum of digits in even positions (2, 2, 8):

   • Sum of even positions: $2 + 2 + 8 = 12$

4. Calculate the difference:

   • Difference = $(7 + x) - 12 = x - 5$

5. Determine the value of $x$ such that the difference is 0 or a multiple of 11:

   • $x - 5 = 0$ ⟹ $x = 5$
   • $x - 5 = 11$ ⟹ $x = 16$ (not possible since $x$ must be a single digit)

   Therefore, $x = 5$.

So, the number 12_268 becomes 125268 when the blank space is filled with 5.

Part ii: 2755_96

1. Identify the positions of the digits:

   • 2 (odd position)
   • 7 (even position)
   • 5 (odd position)
   • 5 (even position)
   • _ (odd position, blank space)
   • 9 (even position)
   • 6 (odd position)

2. Sum of digits in odd positions (2, 5, _, 6):

   • Let the blank space be represented by $y$.
   • Sum of odd positions: $2 + 5 + y + 6 = 13 + y$

3. Sum of digits in even positions (7, 5, 9):

   • Sum of even positions: $7 + 5 + 9 = 21$

4. Calculate the difference:

   • Difference = $(13 + y) - 21 = y - 8$

5. Determine the value of $y$ such that the difference is 0 or a multiple of 11:

   • $y - 8 = 0$ ⟹ $y = 8$
   • $y - 8 = 11$ ⟹ $y = 19$ (not possible since $y$ must be a single digit)

   Therefore, $y = 8$.

So, the number 2755_96 becomes 2755896 when the blank space is filled with 8.

Final Answers:

• i. 12_268 ⟹ 125268
• ii. 2755_96 ⟹ 2755896