Show that two consecutive integers do not share any prime as factor

Question

Can assume a smaller subset of positive integers, as the same result would hold for the bigger set.
Let the two positive integers be $x, x+1$, and one would be even and the other odd. So, there cannot be a common prime factor.
The above approach seems incomplete, although definitely there would be no common factors for an even and an odd number. If there were a simple proof that is rigorous too.

Answer 1

That proof is no proof at all. The number $6$ is even, the number $9$ is odd and they share a common prime factor: $3$.

Here's a proof: if $p$ is prime and $p$ divided both $n$ and $n+1$, then $p$ divides their difference, that is, $p$ divides $1$, which is impossible.