prove the sum of an odd iteger and an even integer is oddSol

prove: the sum of an odd iteger and an even integer is odd

Solution

This follows directly from the definitions:

- Odd integer = 2n + 1, where n is any integer.
- Even integer = 2n, where n is any integer.

You\'ve gotta have firm definitions before you start.

An example: Pick an odd and even integer, say 3 and 12. n\'s obviously different here:

3 = 2*1 + 1 (n = 1)
12 = 2*6 (n = 6)

So you have 3 + 12 = 2*7 + 1, which is odd because of the +1 on the end.

And it doesn\'t matter what odd and even integers you pick: just say that

Odd = 2a + 1
Even = 2b

where a & b are any two integers. Then you have

Odd + Even = 2a + 2b + 1 = 2(a + b) + 1

Since there exists an integer n = a + b for any two integers a & b, the sum of any odd and any even number is odd.