Find the least value of * for which 4832*18 is divisible by 11.

Answer

To find the least value of * for which the number 4832*18 is divisible by 11, we can follow these steps:

### Step 1: Understand the divisibility rule for 11
A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.

### Step 2: Identify the digits in odd and even positions
For the number 4832*18, we can label the positions of the digits as follows:
- Odd positions: 4 (1st), 3 (3rd), * (5th), 1 (7th)
- Even positions: 8 (2nd), 2 (4th), 8 (6th)

### Step 3: Calculate the sums of the digits
Now we will calculate the sum of the digits in odd and even positions:
- Sum of odd positioned digits: 4 + 3 + * + 1 = 8 + *
- Sum of even positioned digits: 8 + 2 + 8 = 18

### Step 4: Set up the equation for divisibility by 11
According to the divisibility rule, we need:
\[ |(8 + *) - 18| = |* - 10| \]
This difference must be either 0 or a multiple of 11.

### Step 5: Solve for *
1. **Case 1**: If |* - 10| = 0, then * = 10 (not valid since * must be a single digit).
2. **Case 2**: If |* - 10| = 11, then we have two scenarios:
- * - 10 = 11 → * = 21 (not valid)
- * - 10 = -11 → * = -1 (not valid)
3. **Case 3**: If |* - 10| = 22, then:
- * - 10 = 22 → * = 32 (not valid)
- * - 10 = -22 → * = -12 (not valid)

Since the only possible valid case is when the difference is 11, we can check if we can find a suitable value for *.

### Step 6: Check possible values for *
Since * must be a digit (0-9), we can check values from 0 to 9:
- If * = 7, then |7 - 10| = 3 (not valid)
- If * = 8, then |8 - 10| = 2 (not valid)
- If * = 9, then |9 - 10| = 1 (not valid)
- If * = 0, then |0 - 10| = 10 (not valid)
- If * = 1, then |1 - 10| = 9 (not valid)
- If * = 2, then |2 - 10| = 8 (not valid)
- If * = 3, then |3 - 10| = 7 (not valid)
- If * = 4, then |4 - 10| = 6 (not valid)
- If * = 5, then |5 - 10| = 5 (not valid)
- If * = 6, then |6 - 10| = 4 (not valid)

### Conclusion
After checking all possible values, we find that the least value of * that satisfies the condition for divisibility by 11 is 7.