To solve the equation \( 13_m + 24_m = 41_m \), we need to understand what the notation \( x_m \) means. The subscript \( m \) indicates that the numbers are in base \( m \). ### Step 1: Convert the numbers from base \( m \) to base 10 1. **Convert \( 13_m \) to base 10**: - The number \( 13_m \) can be expressed in base 10 as: \[ 1 \cdot m^1 + 3 \cdot m^0 = m + 3 \] 2. **Convert \( 24_m \) to base 10**: - The number \( 24_m \) can be expressed in base 10 as: \[ 2 \cdot m^1 + 4 \cdot m^0 = 2m + 4 \] 3. **Convert \( 41_m \) to base 10**: - The number \( 41_m \) can be expressed in base 10 as: \[ 4 \cdot m^1 + 1 \cdot m^0 = 4m + 1 \] ### Step 2: Set up the equation in base 10 Now we can rewrite the original equation \( 13_m + 24_m = 41_m \) in base 10: \[ (m + 3) + (2m + 4) = 4m + 1 \] ### Step 3: Simplify the equation Combine like terms on the left side: \[ m + 3 + 2m + 4 = 4m + 1 \] This simplifies to: \[ 3m + 7 = 4m + 1 \] ### Step 4: Solve for \( m \) Now, we will isolate \( m \): 1. Subtract \( 3m \) from both sides: \[ 7 = 4m - 3m + 1 \] This simplifies to: \[ 7 = m + 1 \] 2. Subtract 1 from both sides: \[ 7 - 1 = m \] Thus, we find: \[ m = 6 \] ### Conclusion The value of \( m \) is **6**. Therefore, the correct option is **B**. ### Explanation of Other Options - **Option A (8)**: If \( m = 8 \), then the left side would be \( 13_8 + 24_8 \) which equals \( 11 + 20 = 31 \) in base 10, while the right side \( 41_8 \) equals \( 33 \) in base 10. This does not satisfy the equation. - **Option C (5)**: If \( m = 5 \), then \( 13_5 + 24_5 \) equals \( 8 + 14 = 22 \) in base 10, while \( 41_5 \) equals \( 21 \) in base 10. This does not satisfy the equation. - **Option D (2)**: If \( m = 2 \), then \( 13_2 + 24_2 \) equals \( 3 + 6 = 9 \) in base 10, while \( 41_2 \) equals \( 5 \) in base 10. This does not satisfy the equation. ### Revision Summary - Convert numbers from base \( m \) to base 10 for clarity. - Combine like terms to simplify the equation. - Isolate \( m \) to find its value. - Check each option to confirm correctness and eliminate incorrect choices.