We have learned that an even integers can be expressed as , where is an integer. Now if we add one to an even integer, it becomes odd. Similarly, if we subtract one from an even integer, it also becomes odd. Therefore, odd integers can be expressed as or , where is an integer. Now, we proceed with the theorem and the proof.
We proved that the sum of two even integers is even. In this post, we show that the sum of two odd integers is even.
The sum of two odd integers is even.
Let and be odd integers. Then, and can be expressed as and respectively, where and are integers. This only means that any odd number can be written as the sum of some even integer and one.
Now, substituting we have .
Since and are integers, is also an integer. It is clear that is an integer is divisible by 2. Therefore, is even. So, the sum of two odd integers is even.