If the digits in the unit and the ten's places of a three digit number are interchanged, a new number is formed, which is greater than the original number by 63. Suppose the digit in the unit place of the original number be x. Then, all the possible values of x are

Options

A. 7,8,9
B. 2,7,9
C. 0,1,2
D. 1,2,8

Answer

To solve the problem step by step, we will denote the three-digit number as \(100z + 10y + x\), where \(z\), \(y\), and \(x\) are the digits in the hundreds, tens, and units places respectively.

### Step 1: Write the original number and the new number after interchanging the digits.
The original number is:
\[ N = 100z + 10y + x \]
When the digits in the unit place (x) and the ten's place (y) are interchanged, the new number becomes:
\[ N' = 100z + 10x + y \]

### Step 2: Set up the equation based on the problem statement.
According to the problem, the new number is greater than the original number by 63:
\[ N' = N + 63 \]
Substituting the expressions for \(N\) and \(N'\):
\[ 100z + 10x + y = (100z + 10y + x) + 63 \]

### Step 3: Simplify the equation.
Now, let's simplify the equation:
\[ 100z + 10x + y = 100z + 10y + x + 63 \]
Subtract \(100z\) from both sides:
\[ 10x + y = 10y + x + 63 \]
Now, rearranging gives:
\[ 10x - x + y - 10y = 63 \]
This simplifies to:
\[ 9x - 9y = 63 \]

### Step 4: Factor out the common term.
Factoring out 9 from the left side:
\[ 9(x - y) = 63 \]
Dividing both sides by 9:
\[ x - y = 7 \]

### Step 5: Determine possible values for \(x\) and \(y\).
Since \(x\) and \(y\) are digits (0-9), we can express \(x\) in terms of \(y\):
\[ x = y + 7 \]
Now, we need to find valid values for \(y\) such that \(x\) remains a digit (0-9).

### Step 6: Find the range for \(y\).
Since \(x\) must be less than or equal to 9:
\[ y + 7 \leq 9 \]
This simplifies to:
\[ y \leq 2 \]
Thus, \(y\) can take values 0, 1, or 2.

### Step 7: Calculate corresponding values of \(x\).
- If \(y = 0\), then \(x = 0 + 7 = 7\)
- If \(y = 1\), then \(x = 1 + 7 = 8\)
- If \(y = 2\), then \(x = 2 + 7 = 9\)

### Conclusion:
The possible values of \(x\) are:
\[ \{7, 8, 9\} \]