-1

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

General Grievance
  • 159
Marcus F
  • 204
  • 1
    Everything looks fine. The first 2 lines are not necessary. You forgot $m_1m_2$ from the line before last. You could also dispense with indices by writing: $m=2p$ and $n=2q$ to start. This wouldn't change the proof, only make it simpler to read. – MasB Apr 25 '20 at 11:32
  • It is special case of $,2\mid n,\Rightarrow,2\mid nm,,$ or, more generally, that multiples are closed under integer scaling - see the linked dupe (more specifically see here, which can't be added to the dupe list since it got no upvotes). There are many similar prior questions. – Bill Dubuque Apr 25 '20 at 18:04
  • why do you put almost all of your text to boldface? – miracle173 Apr 25 '20 at 18:17
  • More generally sets of (common) multiples are closed under addition and scalings - a prototype of a fundamental ring-theoretic structure known as an ideal. – Bill Dubuque Apr 25 '20 at 18:35

1 Answers1

-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.

varpi
  • 587