Evens and Odds

[d0ntc4r3 104]

Evens and Odds

By Gallo

 

 

 

The sum of two even numbers always results in an even number:

 

Let us suppose that we have a random even integer. Now let us give it a name, n. We can say that, because n is even, n is the product of 2 and some other integer k. We can write this as n = 2 * k

 

Now let us suppose we have another even number m, which is also the product of 2 and some other integer q. Then m = 2 * q

 

So if we were to add the numbers n and m (n + m), what we are really doing is adding their products.

 

We can rewrite n + m as 2k + 2q

 

Let us get one certain fact our of the way: the sum of two numbers  with common factors can be rewritten as the common factor multiplied by the sum of the numbers, each divided by their common factor.

 

So our sum 2k + 2q can be rewritten as 2 * (k + q)

 

Fundamentally, the sum of two integers always results in an integer. So let us say that k + q equals some other integer p. Then our summation can rewritten as 2 * p

 

This satisfies our previous definition of even numbers.

 

Therefore, the sum of two even numbers always results in an even number.

 

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The sum of two odd numbers always results in an even number:

 

Let us suppose that we have an odd number named n (and let every number henceforth named be different from the previous set of numbers).

 

Because n is odd, it has the certain property that is the sum of an even number and 1. So let us say we have integer k, then we have n = 2k + 1

 

Now let us say we have odd number m, which is the sum of 2q + 1 (q being another integer).

 

So if we were to add integers n and m, we are really saying 2k + 1 + 2q + 1

 

Let us rearrange this sum to our advantage: 2k + 2q + 2

 

This can be rewritten as 2(k + q + 1)

 

k + q + 1 will sum to an integer, and let us name this integer p. Then we can say (2k + q + 1) = 2p

 

This satisfies our previously established definition of even.

 

Therefore, the sum of two odd numbers always results in an even number.

 

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The sum of an even number and an odd number always results in an odd number:

 

Let us say we have even number n, which is the product of 2 and some integer k. Then, n = 2k

 

Now let us say we have odd number m, which is the sum of 2 multiplied by another integer q and 1. So m = 2q + 1

 

If we were to add n and m, we can say n + m = 2k + 2q + 1

 

We can rewrite this as 2(k + q) + 1

 

k + q will sum to some other integer, and let us name this integer p. Then we can say we have 2p + 1

 

This satisfies our definition of an odd number.

 

Therefore, the sum of an even number and an odd number will always result in an odd number.

Edited May 16, 2021 by d0ntc4r3