If 34P7 is divisible by 11, then what is the value of P?

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to determine the value of \( P \) in the number \( 34P7 \) such that it is divisible by 11. We will use the divisibility rule for 11, which states that 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-by-Step Solution:

1. **Identify the digits and their positions**:
- The number is \( 34P7 \).
- The digits in odd positions are: \( 3 \) (1st position), \( P \) (3rd position), and the digits in even positions are: \( 4 \) (2nd position), \( 7 \) (4th position).

2. **Sum the odd and even positioned digits**:
- Sum of digits in odd positions: \( 3 + P \)
- Sum of digits in even positions: \( 4 + 7 = 11 \)

3. **Apply the divisibility rule**:
- According to the rule, we need to check the difference:
\[ |(3 + P) - 11| = |P - 8| \]
- For \( 34P7 \) to be divisible by 11, \( |P - 8| \) must be either 0 or divisible by 11.

4. **Set up the equations**:
- The first case is when \( P - 8 = 0 \):
\[ P = 8 \]
- The second case is when \( P - 8 = 11 \):
\[ P = 19 \quad (\text{not valid since } P \text{ must be a single digit}) \]
- The third case is when \( P - 8 = -11 \):
\[ P = -3 \quad (\text{not valid since } P \text{ must be a single digit}) \]

5. **Conclusion**:
- The only valid solution is \( P = 8 \).

Thus, the value of \( P \) that makes \( 34P7 \) divisible by 11 is \( \boxed{8} \).