In math class today we started talking about proofs that odd + odd is even. We went over the basic proof (using 2k+1 and equations etc) and I realized that the only reason that this property exists is because the distance between two consecutive numbers divisible by 2 is one. So I wrote an alternate proof of why odd + odd = even and I wanted to share it here and ask if anyone has anything else to say. So my question basically is... did I get anywhere with this?
Proof that the sum of two odd numbers is an even number: An even number is defined as a number that is divisible by 2. An odd number is a number that is not divisible by 2 but is divisible by 1. The reason that two odds are an even is that the difference between odd and even is only 1, and odd numbers are 1 more than even numbers. For example, we have the number 7. 7 is not divisible by 3. However, adding a random number that is not divisible by 3 either will only give you a number that is divisible by 3 half the time (because the gap between two numbers divisible by 3 is 2). Another example, the number 21. 21 is not divisible by 10, and the gap between consecutive numbers divisible by 10 is 9. That means that if a random number is added to 21 that is not divisible by 10 there is a 1/9 chance that the new number is divisible by 10. For odd numbers, the gap between two consecutive numbers divisible by 2 is only 1, which gives a 1/1 chance of creating an even number.