Proof that Square of Even Number Is Even

When we multiply an even number by itself, let see what happens. Let us have some examples.

2 \times 2 = 4
4 \times 4 = 16
6 \times 6 = 36
8 \times 8 = 64
10 \times 10 = 100

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.

Proof:

Let n be an even number.

Then n = 2k for some integer k.  This only means that any even number n can be represented as the product of 2 and some integer k.

Now, n^2 = (2k)^2 = 4k^2.

Notice that 4k^2 = 2(2k^2) and we can represent 2k^2 as t which is an integer making 2(2k^2) = 2t

Therefore, 4k^2 = 2t and clearly 2t is an even integer.

This means that  4k^2, which is the square of an even number is even.

Therefore, if n is even, its square is also even and we are done.

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