if `a_(1)b_(1)c_(1), a_(2)b_(2)c_(2)" and " a_(3)b_(3)c_(3)` are three-digit even natural numbers and `Delta = |{:(c_(1),,a_(1),,b_(1)),(c_(2),,a_(2),,b_(2)),(c_(3),,a_(3),,b_(3)):}|" then " Delta ` is

Options

A. divisible by 2 but not necessarily by 4
B. divisible by 4 but not necessarily by 8
C. divisible by 8
D. none of these

Explanation

To solve the problem, we need to evaluate the determinant \( \Delta = | \begin{pmatrix} c_1 & a_1 & b_1 \\ c_2 & a_2 & b_2 \\ c_3 & a_3 & b_3 \end{pmatrix} | \) where \( a_i, b_i, c_i \) are three-digit even natural numbers.

### Step-by-Step Solution:

1. **Identify the Structure of the Determinant**:
\[ \Delta = | \begin{pmatrix} c_1 & a_1 & b_1 \\ c_2 & a_2 & b_2 \\ c_3 & a_3 & b_3 \end{pmatrix} | \]

2. **Express \( c_i \) as Even Numbers**:
Since \( c_1, c_2, c_3 \) are even natural numbers, we can express them in the form:
\[ c_1 = 2m_1, \quad c_2 = 2m_2, \quad c_3 = 2m_3 \]
where \( m_1, m_2, m_3 \) are integers.

3. **Substitute \( c_i \) in the Determinant**:
Substitute the expressions for \( c_i \) into the determinant:
\[ \Delta = | \begin{pmatrix} 2m_1 & a_1 & b_1 \\ 2m_2 & a_2 & b_2 \\ 2m_3 & a_3 & b_3 \end{pmatrix} | \]

4. **Factor Out the Common Term**:
Since each entry in the first column has a factor of 2, we can factor out 2 from the determinant:
\[ \Delta = 2 \cdot | \begin{pmatrix} m_1 & a_1 & b_1 \\ m_2 & a_2 & b_2 \\ m_3 & a_3 & b_3 \end{pmatrix} | \]

5. **Let the Inner Determinant be \( k \)**:
Let:
\[ k = | \begin{pmatrix} m_1 & a_1 & b_1 \\ m_2 & a_2 & b_2 \\ m_3 & a_3 & b_3 \end{pmatrix} | \]
Then we can express \( \Delta \) as:
\[ \Delta = 2k \]

6. **Conclusion on Divisibility**:
Since \( \Delta = 2k \), it is clear that \( \Delta \) is divisible by 2. However, we cannot conclude that it is divisible by 4 or 8 without additional information about \( k \). Therefore, we can say:
- \( \Delta \) is divisible by 2 but not necessarily by 4.

### Final Answer:
Thus, the value of \( \Delta \) is divisible by 2 but not necessarily by 4.

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