In a three-digit number, the unit digit is twice the tens digit and the tens digit is twice the hundreds digit. The same number is written as 1XY and 1YX in base 8 and base 9 respectively. Find the sum of X and Y in the identical system.

Options

A. 15
B. 7
C. 11
D. Cannot be determined

Answer

To solve the problem step by step, we will follow the conditions given in the question.

### Step 1: Define the digits of the three-digit number
Let the hundreds digit be \( n \). According to the problem:
- The tens digit is twice the hundreds digit: \( \text{Tens digit} = 2n \)
- The units digit is twice the tens digit: \( \text{Units digit} = 2 \times (2n) = 4n \)

### Step 2: Determine the possible values for \( n \)
Since \( n \), \( 2n \), and \( 4n \) must all be single digits (0-9), we can find the possible values for \( n \):
- The maximum value for \( n \) can be 2 because if \( n = 3 \), then \( 4n = 12 \) which is not a single digit.
- If \( n = 0 \), the number would not be a three-digit number.
- Therefore, the possible values for \( n \) are 1 and 2.

### Step 3: Calculate the corresponding digits for each possible \( n \)
1. If \( n = 1 \):
- Hundreds digit = \( 1 \)
- Tens digit = \( 2 \times 1 = 2 \)
- Units digit = \( 4 \times 1 = 4 \)
- The number is \( 124 \).

2. If \( n = 2 \):
- Hundreds digit = \( 2 \)
- Tens digit = \( 2 \times 2 = 4 \)
- Units digit = \( 4 \times 2 = 8 \)
- The number is \( 248 \).

### Step 4: Check which number satisfies the base conditions
The problem states that the number is represented as \( 1XY \) in base 8 and \( 1YX \) in base 9.

- **For the number \( 124 \)**:
- Convert \( 124 \) to base 8:
- \( 124 \div 8 = 15 \) remainder \( 4 \)
- \( 15 \div 8 = 1 \) remainder \( 7 \)
- \( 1 \div 8 = 0 \) remainder \( 1 \)
- Thus, \( 124 \) in base 8 is \( 174 \) (which is \( 1XY \)).
- Convert \( 124 \) to base 9:
- \( 124 \div 9 = 13 \) remainder \( 7 \)
- \( 13 \div 9 = 1 \) remainder \( 4 \)
- \( 1 \div 9 = 0 \) remainder \( 1 \)
- Thus, \( 124 \) in base 9 is \( 147 \) (which is \( 1YX \)).

- **For the number \( 248 \)**:
- Convert \( 248 \) to base 8:
- \( 248 \div 8 = 31 \) remainder \( 0 \)
- \( 31 \div 8 = 3 \) remainder \( 7 \)
- \( 3 \div 8 = 0 \) remainder \( 3 \)
- Thus, \( 248 \) in base 8 is \( 370 \) (which does not match \( 1XY \)).

Since \( 248 \) does not satisfy the base conditions, we will use \( 124 \).

### Step 5: Identify \( X \) and \( Y \)
From the conversions:
- In base 8, \( 1XY = 174 \) gives us \( X = 7 \) and \( Y = 4 \).
- In base 9, \( 1YX = 147 \) confirms \( Y = 4 \) and \( X = 7 \).

### Step 6: Calculate the sum of \( X \) and \( Y \)
Now we find the sum:
\[ X + Y = 7 + 4 = 11 \]

### Final Answer
The sum of \( X \) and \( Y \) is \( 11 \).

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