Understanding Divisibility by 11

The problem requires finding a digit, represented by '*', that makes the number 7235* divisible by 11. We will use the standard divisibility rule for 11 to find this digit.

The Rule for 11

A number is divisible by 11 if the alternating sum of its digits is zero or a multiple of 11. This means we calculate the difference between the sum of digits at odd positions and the sum of digits at even positions (counting from the right).

Applying the Rule to 7235*

Let the number be represented as 7235*, where '*' is the unknown digit.

Sum of digits at odd positions: The digits are *, 3, and 7. Let the unknown digit '*' be represented by the variable $x$. Their sum is $S_{odd} = x + 3 + 7 = x + 10$.

Sum of digits at even positions: The digits are 5 and 2. Their sum is $S_{even} = 5 + 2 = 7$.

Calculating the Unknown Digit (*)

For the number to be divisible by 11, the difference $S_{odd} - S_{even}$ must be equal to 0 or a multiple of 11.

Difference = $(x + 10) - 7 = x + 3$.

We need to find a digit $x$ (where $0 \le x \le 9$) such that $x + 3$ is 0 or a multiple of 11. Let's check the possibilities:

• Case 1: $x + 3 = 0$

  This gives $x = -3$. This is not a valid digit (must be 0-9).

• Case 2: $x + 3 = 11$

  This gives $x = 11 - 3 = 8$. This is a valid digit (0-9).

• Case 3: $x + 3 = 22$

  This gives $x = 22 - 3 = 19$. This is not a valid digit.

• Other cases: For other multiples of 11 (e.g., -11, 33), $x$ would not fall in the 0-9 range.

Therefore, the only possible digit value for '*' (represented by $x$) is 8.

Conclusion

The missing digit '*' in the number 7235* must be 8 for the number to be divisible by 11. The number is 72358.

Verification: We can check the alternating sum: $7 - 2 + 3 - 5 + 8 = 11$. Since 11 is divisible by 11, our result is correct.