Can anyone help me in proving that two consecutive numbers are co-primes?
Let two consecutive numbers are $n$ and $n+1$.
Assume they are not co-primes.
Then $\gcd(n,n+1)=x$, because it can not equal to $1$, $x$ is natural and $x\gt1$
So $x$ divides $n$ as well as $n+1$.
Then $x$ also divides $n+1-n$, by general understanding.
Hence $x$ divides $1$ or $x=1$.
But we have assumed $x\gt 1$.
So by contradiction $n$ & $n+1$ are co-prime.
Is it right or is there any better way to prove that two consecutive numbers are co-prime?