The uploaded image presents a proof of the statement: "An even number plus an odd number is odd." Let's analyze and explain it step by step.

Explanation of the Proof:

1. Definitions:

   • An even number can be expressed as $2n$, where $n$ is any integer.
   • An odd number can be expressed as $2m + 1$, where $m$ is any integer.

2. Adding an even number and an odd number: $(2n) + (2m + 1)$

3. Combine terms: $2n + 2m + 1$

4. Factor out the common term $2$: $2(n + m) + 1$

5. Interpretation:

   • $2(n + m)$ is clearly even because it is divisible by 2.
   • Adding $1$ to an even number results in an odd number.

Thus, the expression simplifies to $2(n + m) + 1$, which is odd.

Conclusion:

The proof shows that adding an even number and an odd number results in an odd number.

Would you like further clarifications or an alternative explanation?

Follow-Up Questions:

1. How do you define even and odd numbers in modular arithmetic?
2. Why does adding 1 to an even number make it odd?
3. Can this result be extended to prove the sum of two odd numbers?
4. How would you prove this statement using examples?
5. Can we generalize this proof to include subtraction (e.g., even - odd)?

Tip:

When proving mathematical properties, always start with definitions to ensure clarity and rigor.