The sum of two odd integers is even.


We show that if x and y are odd integers then x+y is an even integer.

  1. Let x and y be odd integers.

  1. Since x is odd, we know by Definition 1.4 that there is an integer a with x=2a+1.

  1. Likewise, since y is odd, there is an integer b with y=2b +1.

  1. Observe that x+y = (2a +1)+(2b+1)=2a +2b +2=2(a+b+1).

  1. Therefore there is an integer c (namely a+b+1) with x+y=2c.

  1. Therefore by Definition 1.2, 2|(x+y).

  2. Hence (Definition 1.1) x+y is even.