The image shows a proof that the product of two odd numbers is also odd. Here's a breakdown of the explanation in the image:

1. Assume $x$ and $y$ are odd numbers:

   • An odd number can be written in the form $x = 2k + 1$ where $k \in \mathbb{Z}$ (an integer).
   • Similarly, $y = 2k' + 1$ where $k' \in \mathbb{Z}$ (another integer).

2. Product of $x$ and $y$: $x \cdot y = (2k + 1)(2k' + 1)$

3. Expanding the product: $x \cdot y = (2k + 1)(2k' + 1) = 4kk' + 2k + 2k' + 1$

4. Grouping terms: $x \cdot y = 2(2kk' + k + k') + 1$ Here, $2kk' + k + k'$ is an integer, so the product $x \cdot y$ is of the form $2m + 1$, which is odd.

Conclusion:

This demonstrates that the product of two odd numbers is also odd because the result can be expressed as $2m + 1$, which defines an odd number.

Would you like further clarification or details on any step? Here are 5 related questions to deepen your understanding:

1. What is the general form of an even number and how would you approach proving properties of even numbers?
2. How would you prove that the sum of two odd numbers is even?
3. What is the result when an odd number is multiplied by an even number?
4. Can you extend this method to prove properties of numbers modulo other values (e.g., modulo 3)?
5. How would you represent and prove the product of three odd numbers using a similar method?

Tip: Grouping terms properly in algebraic proofs helps in identifying whether the result will be even or odd.