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 will use the divisibility rule for 11. According to this rule, a number is divisible by 11 if the difference between the sum of the digits at the odd positions and the sum of the digits at the even positions is either 0 or divisible by 11.

### Step-by-step Solution:

1. **Identify the positions of the digits:**
- The number is \( 9y8071 \).
- The positions of the digits from left to right are:
- 1st position: 9 (odd)
- 2nd position: \( y \) (even)
- 3rd position: 8 (odd)
- 4th position: 0 (even)
- 5th position: 7 (odd)
- 6th position: 1 (even)

2. **Calculate the sum of the digits at odd positions:**
- Odd positions: 1st (9), 3rd (8), 5th (7)
- Sum of odd positioned digits = \( 9 + 8 + 7 = 24 \)

3. **Calculate the sum of the digits at even positions:**
- Even positions: 2nd (\( y \)), 4th (0), 6th (1)
- Sum of even positioned digits = \( y + 0 + 1 = y + 1 \)

4. **Set up the equation for divisibility by 11:**
- According to the rule, we need to find the difference:
\[ \text{Difference} = \text{Sum of odd positioned digits} - \text{Sum of even positioned digits} \]
\[ \text{Difference} = 24 - (y + 1) = 24 - y - 1 = 23 - y \]

5. **Determine the values of \( y \) such that the difference is divisible by 11:**
- We need \( 23 - y \) to be either 0 or divisible by 11.
- Let's check the possible values of \( y \):
- If \( 23 - y = 0 \) then \( y = 23 \) (not valid since \( y \) must be a single digit)
- If \( 23 - y = 11 \) then \( y = 12 \) (not valid)
- If \( 23 - y = 22 \) then \( y = 1 \) (valid)
- If \( 23 - y = 33 \) then \( y = -10 \) (not valid)

6. **Conclusion:**
- The only valid single-digit value for \( y \) that makes \( 23 - y \) divisible by 11 is \( y = 1 \).

Thus, the smallest value of \( y \) in the number \( 9y8071 \) such that it is divisible by 11 is \( \boxed{1} \).