In how many ways can 729 be expressed as a difference of the square of whole numbers?

Options

A. 4
B. 6
C. 8
D. none of these

Answer

To solve the problem of how many ways 729 can be expressed as a difference of the squares of whole numbers, we can use the identity:

\[ a^2 - b^2 = (a + b)(a - b) \]

This means we can express 729 as a product of two factors, \( (a + b) \) and \( (a - b) \).

### Step-by-step Solution:

1. **Set Up the Equation**:
We start with the equation:
\[ a^2 - b^2 = 729 \]
This can be rewritten using the difference of squares formula:
\[ (a + b)(a - b) = 729 \]

2. **Factor 729**:
Next, we need to find the factors of 729. The prime factorization of 729 is:
\[ 729 = 3^6 \]
The factors of 729 can be found by considering all combinations of the powers of 3.

3. **List the Factors**:
The factors of 729 are:
\[ 1, 3, 9, 27, 81, 243, 729 \]

4. **Form Pairs of Factors**:
Each pair of factors \( (m, n) \) such that \( m \times n = 729 \) can be used to find \( a \) and \( b \). We set:
\[ a + b = m \quad \text{and} \quad a - b = n \]
From these equations, we can solve for \( a \) and \( b \):
\[ a = \frac{m + n}{2} \quad \text{and} \quad b = \frac{m - n}{2} \]

5. **Check for Whole Numbers**:
For \( a \) and \( b \) to be whole numbers, both \( m + n \) and \( m - n \) must be even. This means \( m \) and \( n \) must either both be odd or both be even.

6. **Identify Valid Factor Pairs**:
The factor pairs of 729 are:
- \( (1, 729) \)
- \( (3, 243) \)
- \( (9, 81) \)
- \( (27, 27) \)

Now we check which pairs yield even sums and differences:
- \( (1, 729) \): \( 1 + 729 = 730 \) (even), \( 1 - 729 = -728 \) (not valid)
- \( (3, 243) \): \( 3 + 243 = 246 \) (even), \( 3 - 243 = -240 \) (not valid)
- \( (9, 81) \): \( 9 + 81 = 90 \) (even), \( 9 - 81 = -72 \) (not valid)
- \( (27, 27) \): \( 27 + 27 = 54 \) (even), \( 27 - 27 = 0 \) (valid)

7. **Count Valid Pairs**:
The only valid pair that results in whole numbers is \( (27, 27) \). Thus, there is only **1 way** to express 729 as a difference of squares of whole numbers.

### Final Answer:
The number of ways to express 729 as a difference of the squares of whole numbers is **1**.