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The problem asks us to prove the following statement using either **proof by contraposition** or **proof by contradiction**:

> Suppose \( a \in \mathbb{Z} \). If \( a^2 - 3a + 5 \) is even, then \( a \) is odd.

### Approach: Proof by Contraposition
We will proceed by proving the contrapositive statement:
- The contrapositive of the original statement is: "If \( a \) is even, then \( a^2 - 3a + 5 \) is odd."

#### Step-by-Step Proof:
1. **Assume \( a \) is even**:  
   If \( a \) is even, we can express \( a \) as \( a = 2k \) for some integer \( k \in \mathbb{Z} \).

2. **Substitute \( a = 2k \) into \( a^2 - 3a + 5 \)**:  
   \[
   a^2 - 3a + 5 = (2k)^2 - 3(2k) + 5
   \]
   Simplifying this expression:
   \[
   = 4k^2 - 6k + 5
   \]
   
3. **Analyze the parity of \( 4k^2 - 6k + 5 \)**:
   - \( 4k^2 \) is clearly even because it is a multiple of 4.
   - \( 6k \) is also even because it is a multiple of 2.
   - So, \( 4k^2 - 6k \) is the sum of two even numbers, which is even.
   - Finally, adding 5 (an odd number) to an even number results in an odd number.

   Therefore, \( a^2 - 3a + 5 = 4k^2 - 6k + 5 \) is odd.

4. **Conclusion**:  
   We have shown that if \( a \) is even, then \( a^2 - 3a + 5 \) is odd. This is the contrapositive of the original statement, so by the **law of contrapositive**, the original statement is true:  
   - If \( a^2 - 3a + 5 \) is even, then \( a \) must be odd.

Thus, the proof is complete.

Would you like more details on any part of this explanation, or have any questions?

### Related Questions:
1. How does proof by contraposition differ from direct proof?
2. Can you apply proof by contradiction to this problem instead of contraposition?
3. Why does the parity of \( a^2 - 3a + 5 \) depend on whether \( a \) is even or odd?
4. What are the properties of even and odd numbers that make such proofs possible?
5. How can you generalize this proof technique to other quadratic expressions?

**Tip**: Contrapositive proofs are often simpler when dealing with even and odd properties because they allow you to assume a more manageable hypothesis (such as "even" instead of "odd").

Suppose a ∈ Z. If a^2 - 3a + 5 is even, then a is odd.

Learn how to prove that if a^2 - 3a + 5 is even, then a is odd using proof by contraposition. This problem involves properties of even and odd numbers and algebraic expressions, suitable for Grades 11-12 or university-level math students.

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