I have been asked to prove the following and am having difficulty:
If $m$ and $n$ are even integers, then so are $m+n$ and $mn$.
My professor has hinted to us to use the definition of an even integer in our proof.
This is my proof for $m+n$ thus far:
- Even integers are defined as being divisible by 2.
- So, if $m$, is even, using this definition we can rewrite this as $m=2j$.
- Similarly, we can rewrite $n$ as $n=2k$.
- We now have $m+n=2k+2j$.
- Distributivity then allows us to write $2j+2k=2(j+k)$
- We now have that $m+n=2(j+k)$.
- I now use associativity to create $m+n=(j+k)2$
Next, the definition of divisibility states that 'When $m$ and $n$ are integers, we say $m$ is divisible by $n$ if there exists $j∈ Z$ such that $m=jn$.
This allows me to conclude that, since all even numbers are divisible by 2, that $m+n$ must be an even number.
I am unsure about whether or not my last two steps are correct. Our teacher has hinted to us to use the definition of divisibility, but I am having trouble wrapping my head around a. how to use it, and b. how it is accurate to do so?
Any advice would be much appreciated!