Understanding the Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places (counted from right or left) is either 0 or a multiple of 11.

Applying the Rule to 67843A2

The given seven-digit number is 67843A2.

• Digits at odd places (1st, 3rd, 5th, 7th): 6, 8, 3, 2
• Digits at even places (2nd, 4th, 6th): 7, 4, A

Calculating the Sums and Difference

1. Sum of digits at odd places: $ S_{odd} = 6 + 8 + 3 + 2 = 19 $
2. Sum of digits at even places: $ S_{even} = 7 + 4 + A = 11 + A $
3. Difference: $ Difference = S_{odd} - S_{even} = 19 - (11 + A) = 19 - 11 - A = 8 - A $

Finding the Value of A

For the number 67843A2 to be divisible by 11, the difference ($8 - A$) must be 0 or a multiple of 11. Since A is a single digit (0 to 9):

• If $8 - A = 0$, then $A = 8$. This is a valid single digit.
• If $8 - A = 11$, then $A = 8 - 11 = -3$. This is not a valid digit.
• If $8 - A = -11$, then $A = 8 + 11 = 19$. This is not a valid digit.

The only possible value for A that satisfies the condition is 8.