If 54k31m26 is divisible by 11 then find the maximum value of k + m.

Options

A. 12
B. 10
C. 14
D. 13

Answer

To solve the problem of finding the maximum value of \( k + m \) such that the number \( 54k31m26 \) is divisible by 11, we will follow the divisibility rule for 11. According to this rule, 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 positions of the digits:**
The digits of the number \( 54k31m26 \) can be indexed as follows:
- Odd positions: 5 (1st), k (3rd), 1 (5th), 2 (7th) → Sum = \( 5 + k + 1 + 2 = 8 + k \)
- Even positions: 4 (2nd), 3 (4th), m (6th), 6 (8th) → Sum = \( 4 + 3 + m + 6 = 13 + m \)

2. **Set up the equation for divisibility by 11:**
We need to find the difference between the sums:
\[ (8 + k) - (13 + m) = k - m - 5 \]
For the number to be divisible by 11, \( k - m - 5 \) must be either 0 or a multiple of 11. This gives us two cases to consider:
- Case 1: \( k - m - 5 = 0 \) → \( k - m = 5 \)
- Case 2: \( k - m - 5 = 11 \) → \( k - m = 16 \) (not possible since \( k \) and \( m \) are single digits)

3. **Solve for \( k \) and \( m \) in Case 1:**
From \( k - m = 5 \), we can express \( k \) in terms of \( m \):
\[ k = m + 5 \]
Since \( k \) and \( m \) must be single-digit numbers (0 to 9), the maximum value for \( m \) can be calculated:
- If \( m = 4 \), then \( k = 4 + 5 = 9 \) (valid)
- If \( m = 5 \), then \( k = 5 + 5 = 10 \) (invalid)

Thus, the only valid pair is \( m = 4 \) and \( k = 9 \).

4. **Calculate \( k + m \):**
\[ k + m = 9 + 4 = 13 \]

5. **Conclusion:**
The maximum value of \( k + m \) is \( 13 \).