Which of the following cannot be the difference between a two-digit number and the number obtained by interchanging the digits?

Options

A. 72
B. 36
C. 54
D. 48

Answer

To solve the problem of determining which of the given options cannot be the difference between a two-digit number and the number obtained by interchanging its digits, we can follow these steps:

### Step-by-Step Solution:

1. **Representation of a Two-Digit Number:**
Let the two-digit number be represented as \(XY\), where \(X\) is the digit in the tens place and \(Y\) is the digit in the units place. The numerical value of this number can be expressed as:
\[ 10X + Y \]

2. **Interchanging the Digits:**
When the digits are interchanged, the new number becomes \(YX\). The numerical value of this interchanged number is:
\[ 10Y + X \]

3. **Finding the Difference:**
We need to find the difference between the original number and the interchanged number:
\[ \text{Difference} = (10X + Y) - (10Y + X) \]
Simplifying this expression, we get:
\[ \text{Difference} = 10X + Y - 10Y - X = 9X - 9Y = 9(X - Y) \]

4. **Understanding the Result:**
The difference \(9(X - Y)\) indicates that the result will always be a multiple of 9 since \(X\) and \(Y\) are digits (0-9). The possible values for \(X - Y\) can range from \(-9\) to \(9\), leading to possible differences of:
\[ 9(-9), 9(-8), \ldots, 9(8), 9(9) \]
which translates to:
\[ -81, -72, -63, -54, -45, -36, -27, -18, -9, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81 \]

5. **Identifying Non-Multiples of 9:**
Since the difference must be a multiple of 9, we can now evaluate the options provided in the question. We need to identify which of the options is not a multiple of 9.

6. **Checking Each Option:**
- Option 1: 9 (multiple of 9)
- Option 2: 18 (multiple of 9)
- Option 3: 72 (multiple of 9)
- Option 4: 70 (not a multiple of 9)

7. **Conclusion:**
The only option that cannot be the difference between a two-digit number and the number obtained by interchanging its digits is:
\[ \text{Option 4: 70} \]