Understanding the Divisibility Rule for 11

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

Applying the Rule to 93248x6

Let's consider the number 93248x6. We will count the positions from the left:

• Digits at odd positions (1st, 3rd, 5th, 7th): 9, 2, 8, 6
• Digits at even positions (2nd, 4th, 6th): 3, 4, x

Calculating the Sums

1. Sum of digits at odd places: $9 + 2 + 8 + 6 = 25$
2. Sum of digits at even places: $3 + 4 + x = 7 + x$

Solving for x

According to the divisibility rule for 11, the difference between these sums must be 0 or a multiple of 11.

Difference = (Sum of odd places) - (Sum of even places)

Where 'k' is an integer (0, ±1, ±2, ...).

Since 'x' must be a single digit (from 0 to 9):

• If $k = 0$, then $18 - x = 0 \implies x = 18$ (Not possible as x is a single digit).
• If $k = 1$, then $18 - x = 11 \implies x = 18 - 11 \implies x = 7$ (This is possible).
• If $k = 2$, then $18 - x = 22 \implies x = 18 - 22 \implies x = -4$ (Not possible).
• If $k = -1$, then $18 - x = -11 \implies x = 18 + 11 \implies x = 29$ (Not possible).

The only possible value for the digit x is 7.

Conclusion

The digit x must be 7 for the number 93248x6 to be divisible by 11.