# If $n$ is a natural number, then what are the possible values(s) of gcd $(n+1, 2-n)$?

If $n$ is a natural number, then what are the possible values(s) of gcd $(n+1, 2-n)$?

There are six options: $1,2,3,4,5,6$

I think the answer is $1$

• Consider $n=2$. – Brian M. Scott Dec 11 '13 at 21:08
• They are asking what the possible options are, meaning that you are choosing which of those $6$ values is possible. There will be multiple answers. – Cameron Buie Dec 11 '13 at 21:08

$gcd(a,b)=gcd(a,a+b)$
$gcd (n+1,2−n)=gcd(n+1,3)\in\{1,3\}$
If $p|n+1$ and $p|2-n$, then $p|n+1+2-n=3$, thus $p=1$ or $p=3$.
The case $p=3$ can happen with $n=2+3k$.
For other value of $n$, the gcd is of course 1.