Replace * by the largest digit so that the number 40643* is divisible by 11

Answer

To solve the problem of replacing * in the number 40643* so that it is divisible by 11, we will 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 divisible by 11.

### Step 2: Identify the digits in the number 40643*
The digits in the number are:
- 4 (1st position)
- 0 (2nd position)
- 6 (3rd position)
- 4 (4th position)
- 3 (5th position)
- * (6th position)

### Step 3: Calculate the sum of the digits in odd positions
The odd positioned digits are: 4 (1st), 6 (3rd), and 3 (5th).
Sum of odd positioned digits = 4 + 6 + 3 = 13

### Step 4: Calculate the sum of the digits in even positions
The even positioned digits are: 0 (2nd), 4 (4th), and * (6th).
Sum of even positioned digits = 0 + 4 + * = 4 + *

### Step 5: Set up the equation for divisibility by 11
According to the rule, we need to find the value of * such that:
| (Sum of odd positioned digits) - (Sum of even positioned digits) | = 0 or is divisible by 11.

This gives us the equation:
| 13 - (4 + *) | = 0 or divisible by 11

### Step 6: Simplify the equation
This simplifies to:
| 13 - 4 - * | = | 9 - * |

### Step 7: Determine the possible values for *
We want | 9 - * | to be 0 or divisible by 11. The possible values for * can be calculated as follows:
1. If 9 - * = 0, then * = 9.
2. If 9 - * = 11, then * = -2 (not valid since * must be a digit).
3. If 9 - * = -11, then * = 20 (not valid since * must be a digit).

The only valid solution is when * = 9.

### Step 8: Conclusion
Thus, the largest digit that can replace * in the number 40643* to make it divisible by 11 is 9.

### Final Answer
* = 9

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