## Examples 1.1.5(a):

Prove that when two even integers are multiplied, the result is an even integer,
and when two odd integers are multiplied, the result is an odd integer.

To prove this we first need to know what exactly an even and odd integer is:
- an integer
*x*is even if*x = 2n*for some integer*n* - an integer
*x*is odd if*x = 2n + 1*for some integer*n*

*x*and

*y*two even numbers. Then

*x = 2n*for some integer*n**y = 2m*for some integer*m*

*xy = (2n)(2m) = 4 nm = 2 (2nm) = 2 k*

*k = 2nm*. Hence,

*xy*is again even.

If *x* and *y* are two odd numbers, then

*x = 2n + 1*for some integer*n**y = 2m + 1*for some integer*m*

*xy = (2n+1)(2m+1) = 4nm + 2(n + m) + 1 = 2 (2nm + n + m) + 1 = 2k + 1*

*k = 2nm + n + m*. Hence,

*xy*is again odd.