When we multiply an even number by itself, let see what happens. Let us have some examples.
Notice that if we multiply even numbers by itself, the product is even. Now how do we prove that this is always true?
Theorem: The square of an even number is even.
Let be an even number.
Then for some integer . This only means that any even number can be represented as the product of and some integer .
Notice that and we can represent as which is an integer making
Therefore, and clearly is an even integer.
This means that , which is the square of an even number is even.
Therefore, if is even, its square is also even and we are done.