Question

Prove that the sum of two odd numbers is even.

Answer 1

To prove that two odd numbers add up to an even number, represent odd numbers as 2n + 1 and add two such terms. The result, 2[(n + m) + 1], is a multiple of 2, signifying an even sum.

Step-by-step explanation:

To prove that the sum of two odd numbers is even, we start by defining odd numbers. An odd number can be represented as 2n + 1, where n is an integer. When we add two odd numbers, we get (2n + 1) + (2m + 1), where n and m are integers.

Let's simplify this expression:

1. First, add the integers with the same coefficient: 2n + 2m. This is equivalent to 2(n + m).
2. Now, add the ones: 1 + 1, which equals 2.
3. Combine both results: 2(n + m) + 2. This can be factored as 2[(n + m) + 1].
4. The final expression 2[(n + m) + 1] shows that the sum is a multiple of 2, which confirms that it is an even number.

Therefore, it is proven that the sum of any two odd numbers will always be an even number.