The smallest value of y in the number 9y8071, so that it is divisible by 11 is

Options

A. 2
B. 3
C. 1
D. 0

Answer

To find the smallest value of \( y \) in the number \( 9y8071 \) such that it is divisible by 11, we can follow these steps:

### Step 1: Identify the positions of the digits
The number \( 9y8071 \) has the following positions:
- 1st place (odd): 9
- 2nd place (even): \( y \)
- 3rd place (odd): 8
- 4th place (even): 0
- 5th place (odd): 7
- 6th place (even): 1

### Step 2: Calculate the sum of the digits in odd and even positions
- **Sum of odd position digits**: \( 9 + 8 + 7 = 24 \)
- **Sum of even position digits**: \( y + 0 + 1 = y + 1 \)

### Step 3: Find the difference between the sums
The difference between the sum of the odd position digits and the sum of the even position digits is:
\[ \text{Difference} = |(24) - (y + 1)| = |23 - y| \]

### Step 4: Apply the divisibility rule for 11
For the number to be divisible by 11, the difference calculated above must be either 0 or a multiple of 11. Therefore, we can set up the following conditions:
\[ 23 - y = 0 \quad \text{or} \quad 23 - y = 11 \quad \text{or} \quad 23 - y = -11 \]

### Step 5: Solve the equations
1. **For \( 23 - y = 0 \)**:
\[ y = 23 \quad \text{(not valid since } y \text{ must be a single digit)} \]

2. **For \( 23 - y = 11 \)**:
\[ 23 - 11 = y \implies y = 12 \quad \text{(not valid since } y \text{ must be a single digit)} \]

3. **For \( 23 - y = -11 \)**:
\[ 23 + 11 = y \implies y = 34 \quad \text{(not valid since } y \text{ must be a single digit)} \]

### Step 6: Check possible values of \( y \)
Since \( y \) must be a single digit (0-9), we can check values from 0 to 9 to find the smallest valid \( y \) such that \( |23 - y| \) is a multiple of 11.

- **If \( y = 0 \)**: \( |23 - 0| = 23 \) (not a multiple of 11)
- **If \( y = 1 \)**: \( |23 - 1| = 22 \) (not a multiple of 11)
- **If \( y = 2 \)**: \( |23 - 2| = 21 \) (not a multiple of 11)
- **If \( y = 3 \)**: \( |23 - 3| = 20 \) (not a multiple of 11)
- **If \( y = 4 \)**: \( |23 - 4| = 19 \) (not a multiple of 11)
- **If \( y = 5 \)**: \( |23 - 5| = 18 \) (not a multiple of 11)
- **If \( y = 6 \)**: \( |23 - 6| = 17 \) (not a multiple of 11)
- **If \( y = 7 \)**: \( |23 - 7| = 16 \) (not a multiple of 11)
- **If \( y = 8 \)**: \( |23 - 8| = 15 \) (not a multiple of 11)
- **If \( y = 9 \)**: \( |23 - 9| = 14 \) (not a multiple of 11)

### Conclusion
After checking all possible values, the smallest valid \( y \) that satisfies the divisibility rule for 11 is \( y = 1 \).

Thus, the answer is:
\[ \text{The smallest value of } y \text{ is } 1. \]