How many two-digit numbers exist such that the difference of the squares of its digits is 24?

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem of how many two-digit numbers exist such that the difference of the squares of its digits is 24, we can follow these steps:

### Step 1: Define the digits of the two-digit number
Let the two-digit number be represented as \( xy \), where \( x \) is the tens digit and \( y \) is the units digit. Since \( x \) and \( y \) are digits, \( x \) can range from 1 to 9 (as it cannot be 0 in a two-digit number) and \( y \) can range from 0 to 9.

### Step 2: Set up the equation based on the problem statement
According to the problem, we need to find the digits such that the difference of the squares of the digits is 24. This can be expressed mathematically as:
\[ x^2 - y^2 = 24 \]

### Step 3: Factor the equation
Using the difference of squares factorization, we can rewrite the equation:
\[ (x - y)(x + y) = 24 \]

### Step 4: Find the factor pairs of 24
Next, we need to find the pairs of integers \((a, b)\) such that \( ab = 24 \). The factor pairs of 24 are:
- (1, 24)
- (2, 12)
- (3, 8)
- (4, 6)

### Step 5: Solve for \( x \) and \( y \)
For each factor pair \((a, b)\), we can set:
\[ x - y = a \quad \text{and} \quad x + y = b \]
From these two equations, we can solve for \( x \) and \( y \):
\[ x = \frac{(x - y) + (x + y)}{2} = \frac{a + b}{2} \]
\[ y = \frac{(x + y) - (x - y)}{2} = \frac{b - a}{2} \]

### Step 6: Check each factor pair
1. **For (1, 24)**:
- \( x = \frac{1 + 24}{2} = 12.5 \) (not valid)

2. **For (2, 12)**:
- \( x = \frac{2 + 12}{2} = 7 \)
- \( y = \frac{12 - 2}{2} = 5 \)
- Valid pair: \( (7, 5) \) → Number: 75

3. **For (3, 8)**:
- \( x = \frac{3 + 8}{2} = 5.5 \) (not valid)

4. **For (4, 6)**:
- \( x = \frac{4 + 6}{2} = 5 \)
- \( y = \frac{6 - 4}{2} = 1 \)
- Valid pair: \( (5, 1) \) → Number: 51

### Step 7: Consider reverse pairs
From the valid pairs found:
- From \( (7, 5) \) we have 75
- From \( (5, 1) \) we have 51

We also need to consider the reverse of these pairs:
- \( (5, 7) \) → Number: 57
- \( (1, 5) \) → Number: 15

### Step 8: List all valid two-digit numbers
The valid two-digit numbers are:
- 15
- 51
- 57
- 75

### Conclusion
Thus, there are a total of 4 two-digit numbers such that the difference of the squares of their digits is 24.

**Final Answer: 4**