Use that an odd integer can be written as 2n+1 where n is an integer.
The difference of two odd integers is an even integer.
Practice makes perfect
Let's first look as three examples to see if we can spot any pattern.
5-3&=2
11-5&=6
17-9&=8We notice that each difference is an even integer. We will now show that this is true for all odd integers. An odd integer can be written as 2n+1, where n is an integer. We can write the second odd integer as 2m+1, where m is an integer. We can write the difference of these two integers as follows.
2n+1 -(2m+1)
Let's simplify and rewrite this.
We can write the difference of the two odd integers as 2(n-m). Since (n-m) is an integer we can draw the conclusion that 2(n-m) is an even integer and that the difference of any two odd integers is an even integer.
