Krdytkyu

Prove that $gcd(n,n+2)=1$ if $n$ is odd and $gcd(n,n+2)=2$ if $n$ is even.

My Try:

So, first I took some $k$ to be even then $k$ is not the common divisor of $n$ and $n+2$.

If $k|n$ and $k|n+2$ as $k$ is even $implies 2|k$

$2|n$ and $2|n+2$ is not possible.

Is my above attempt correct? Are there any better ways to prove the above?

An easy way to prove it would be:

Part 1)

Suppose $n=2k+1$ since $n$ is odd, therefore $n+2=2k+3$
Now if we want to find their gcd we would get that $$gcd(n,n+2)=gcd(2k+1,2k+3)=gcd(2k+1,2k+3-(2k+1))=gcd(2k+1,2)$$ But $2k+1$ is odd while $2$ is even, therefore their gcd will have to be $1$. Meaning that $gcd(n, n+2)=1$ if $n$ is odd.

Part 2)

Now we can suppose $n=2k$, therefore $n+2=2k+2$
Now if we want to find their gcd we would get that $$gcd(n,n+2)=gcd(2k,2k+2)=gcd(2k,2k+2-(2k))=gcd(2k,2)$$
Now since $2k$ is even, that means $2$ can be a common divisor and it would be the largest as we know that $gcd(a, b)le a, b$. This gives us that $gcd(n, n+2)=2$ if $n$ is even.

I did not quite understand your attempted solution.

So, first I took some $k$ to be even then $k$ is not the common divisor of $n$ and $n+2$.

Are you saying that if $k$ is even, then $k$ is not the common divisor of $n$ and $n+2$?
That won't be true unless $n$ is odd. Maybe you meant to say "We consider the case that $n$ is odd first."?

Additionally, this statement requires justification. Did you mean to say, "First I will show that for any even number $k$, $k$ is not the common divisor of $n$ and $n+2$."?

If $k|n$ and $k|n+2$ as $k$ is even $implies 2|k$

I think you meant to say: "Since $k$ is even, $2 mid k$. Additionally, suppose towards contradiction that $k mid n$ and $k mid n+2$ (we will show that this is impossible). Then also $2 mid n$ and $2 mid n+2$."

$2|n$ and $2|n+2$ is not possible.

I would not say it's "not possible". Do you mean that it contradicts the fact that $n$ is odd, which we assumed earlier?

If you're doing a proof by contradiction, always say so, and end with a contradiction!

Is my above attempt correct? Are there any better ways to prove the above?

There is a problem with your proof. First, it seems like you only considered the case where $n$ is odd. Second, it seems like you just showed that the common divisor of $n$ and $n+2$ cannot be even. But, what if the common divisor of $n$ and $n+2$ is $3$? What if it is $5$? You need to show that it is $1$, not just that it is not even.

I would suggest you try to write down your ideas more carefully. Once you can write down the ideas carefully, it may help you write a correct solution. Do not allow yourself to make any jumps in the argument that are not perfectly logical and correct.

There are some areas left vague, the part proving $k$ is not a greatest common divisor of $n$ and $n+2$ (actually this is false, let $k=2$) and showing $2|n$ and $2|n+2$ is impossible.

Part 1: you need to add that $k>2$ or else the statement is false, and show why the statement is true for $k>2$. You can show that for $nequiv 0mod k$ and $n+2equiv 0mod k$, we can convert the statements to $0equiv -nmod k$ and $2equiv -nmod k$ and since $-nequiv 0mod k$, the statements become $2equiv0mod k$ (we can ignore $0equiv0mod k$ since that is true for all $k$ and is redundant). We can see that the above statement is not true for $k>2$, so the statement is false.

Part 2: Let us define $m$ such that $m=n-1$ if $n$ is odd. Since $2|m$, $nequiv1mod2$ and $n+2equiv3equiv1mod2$. So $2nmid n$ and $2nmid n+2$ for odd $n$. Since Part $1$, when generalized to all values of $k$, involves $nequiv0mod2$, it shows $n+2equiv0mod2$.