Proof of even numbers, is my proof correct?

Question

i have gotten a question that sounds like this:

• An even number, n, is characterized in that it can be written as n = 2 * m, where m is an integer. Proof (or argument as to why), once you multiply two even numbers with each other, you get an even number.

So far i have done this , but im not sure if it is correct or if my explanation is thorough enough:

n_1 * n_2

n_1 * n_2 is equal, therefore

n_1 = 2m_1

n_2 = 2m_2

thus:

n_1 * n_2 = 2m_1 * 2m_2 = 2 * 2

Since n_1 * n_2 = 2 * 2 m_1 m_2, n_1 * n_2 is an even number

Answer 1

Your idea is pretty much correct, perhaps the (re)writing here is in better detail:

Let $n_1$ and $n_2$ be two even numbers such that $$n_1 = 2m_1$$ $$n_2 = 2m_2$$ for $m_1,m_2\in\mathbb{Z}$. Then, we have $$n_1\cdot n_2 = 2m_1\cdot 2m_2 = 2\cdot(2m_1m_2).$$ Now, let $m=2m_1m_2$ and observe that $m\in\mathbb{Z}$. Hence, we have $$n_1\cdot n_2 = 2m.$$ Thus, by the definition of an even integer, $n_1\cdot n_2$ is even.

Also, as a small reminder, it's generally easier for readers / answerers to read your question if you use math-type / LaTex formatting.