To prove the statement "An even number plus an odd number is odd," let's analyze the provided image and the steps taken:

Proof Explanation:

1. Representation of Even and Odd Numbers:

   • Any even number can be written as $2n$, where $n$ is an integer.
   • Any odd number can be written as $2m + 1$, where $m$ is an integer.

2. Adding the Even and Odd Numbers: $(2n) + (2m + 1)$

3. Combine Like Terms: $2n + 2m + 1$

4. Factor Out $2$ From $2n + 2m$: $2(n + m) + 1$

   • Here, $n + m$ is still an integer because the sum of two integers is an integer.

5. Conclusion:

   • The expression $2(n + m) + 1$ has the form of an odd number because it consists of an even term $2(n + m)$ plus 1.
   • Thus, the sum of an even number and an odd number is always odd.

Would you like a deeper breakdown or additional examples to further solidify this concept?

Related Questions:

1. Can we extend this proof to subtraction (e.g., an even number minus an odd number)?
2. How does the addition of two even numbers differ from this case?
3. Can this result be verified using modular arithmetic?
4. Is the sum of two odd numbers even? Why?
5. Can this proof be applied in other fields, such as computer science or logic?

Tip:

When working with proofs involving even and odd numbers, modular arithmetic (mod 2) often provides a quick and elegant way to verify results.