Replace * by the smallest digit so that:
43*439 is divisible by 11

Answer

To determine the smallest digit that can replace * in the number 43*439 such that the entire number is divisible by 11, we can follow these steps:

### Step 1: Understand the divisibility rule for 11
The rule states that 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.

### Step 2: Identify the positions of the digits
For the number 43*439:
- Odd positions: 4 (1st), * (3rd), 3 (5th)
- Even positions: 3 (2nd), 4 (4th), 9 (6th)

### Step 3: Calculate the sums
- Sum of odd position digits: 4 + * + 3 = 7 + *
- Sum of even position digits: 3 + 4 + 9 = 16

### Step 4: Set up the equation
According to the divisibility rule:
\[ (7 + *) - 16 = * - 9 \]
This difference must be a multiple of 11.

### Step 5: Solve for *
We need to find the smallest digit (0 to 9) that makes \( * - 9 \) a multiple of 11.

1. If we let \( * = 0 \):
\[ 0 - 9 = -9 \] (not a multiple of 11)

2. If we let \( * = 1 \):
\[ 1 - 9 = -8 \] (not a multiple of 11)

3. If we let \( * = 2 \):
\[ 2 - 9 = -7 \] (not a multiple of 11)

4. If we let \( * = 3 \):
\[ 3 - 9 = -6 \] (not a multiple of 11)

5. If we let \( * = 4 \):
\[ 4 - 9 = -5 \] (not a multiple of 11)

6. If we let \( * = 5 \):
\[ 5 - 9 = -4 \] (not a multiple of 11)

7. If we let \( * = 6 \):
\[ 6 - 9 = -3 \] (not a multiple of 11)

8. If we let \( * = 7 \):
\[ 7 - 9 = -2 \] (not a multiple of 11)

9. If we let \( * = 8 \):
\[ 8 - 9 = -1 \] (not a multiple of 11)

10. If we let \( * = 9 \):
\[ 9 - 9 = 0 \] (which is a multiple of 11)

### Conclusion
The smallest digit that can replace * to make 43439 divisible by 11 is **9**.