To solve the equation \(P344_{6} - 23P2_{6} = 2PP2_{6}\), we need to first convert all the numbers from base 6 to base 10. This will allow us to perform the arithmetic operations more easily. ### Step 1: Convert each number from base 6 to base 10 1. **Convert \(P344_{6}\)**: \[ P344_{6} = P \cdot 6^3 + 3 \cdot 6^2 + 4 \cdot 6^1 + 4 \cdot 6^0 \] \[ = P \cdot 216 + 3 \cdot 36 + 4 \cdot 6 + 4 \cdot 1 \] \[ = 216P + 108 + 24 + 4 = 216P + 136 \] 2. **Convert \(23P2_{6}\)**: \[ 23P2_{6} = 2 \cdot 6^2 + 3 \cdot 6^1 + P \cdot 6^0 \] \[ = 2 \cdot 36 + 3 \cdot 6 + P \cdot 1 \] \[ = 72 + 18 + P = 90 + P \] 3. **Convert \(2PP2_{6}\)**: \[ 2PP2_{6} = 2 \cdot 6^3 + P \cdot 6^2 + P \cdot 6^1 + 2 \cdot 6^0 \] \[ = 2 \cdot 216 + P \cdot 36 + P \cdot 6 + 2 \cdot 1 \] \[ = 432 + 36P + 6P + 2 = 432 + 42P + 2 = 434 + 42P \] ### Step 2: Set up the equation Now we can substitute these values into the original equation: \[ (216P + 136) - (90 + P) = 434 + 42P \] ### Step 3: Simplify the equation 1. Distributing the negative sign: \[ 216P + 136 - 90 - P = 434 + 42P \] \[ 215P + 46 = 434 + 42P \] 2. Rearranging the equation: \[ 215P - 42P = 434 - 46 \] \[ 173P = 388 \] 3. Solving for \(P\): \[ P = \frac{388}{173} \approx 2.24 \] Since \(P\) must be a digit in base 6, it can only take integer values from 0 to 5. The only integer value that fits this equation is \(P = 2\). ### Step 4: Verify the solution To ensure that \(P = 2\) is correct, we can substitute it back into the original numbers and check if the equation holds true. 1. **Substituting \(P = 2\)**: - \(P344_{6} = 2344_{6}\) - \(23P2_{6} = 2322_{6}\) - \(2PP2_{6} = 2222_{6}\) 2. **Convert these back to base 10**: - \(2344_{6} = 2 \cdot 216 + 3 \cdot 36 + 4 \cdot 6 + 4 = 432 + 108 + 24 + 4 = 568\) - \(2322_{6} = 2 \cdot 36 + 3 \cdot 6 + 2 = 72 + 18 + 2 = 92\) - \(2222_{6} = 2 \cdot 216 + 2 \cdot 36 + 2 \cdot 6 + 2 = 432 + 72 + 12 + 2 = 518\) 3. **Check the equation**: \[ 568 - 92 = 518 \] This confirms that the equation holds true. ### Conclusion The correct answer is **A. 2**. ### Summary - Convert numbers from base 6 to base 10 for easier calculations. - Rearrange and simplify the equation to isolate the variable. - Verify the solution by substituting back into the original equation. - Remember that \(P\) must be a valid digit in base 6 (0-5).