what is the GCD of n and n+1
It is $\displaystyle 1$, obviously ... can $\displaystyle n + 1$ possibly divide $\displaystyle n$ ? No. You can investigate this fraction :
$\displaystyle \frac{\ n + 1}{n}$
And saying that two consecutive numbers have a common factor (except 1) is saying that :
$\displaystyle (n + 1) mod n = 0$
Which is impossible.