(a) Prove that consecutive integers have no prime factors in common.
(b) Prove that $\gcd (n, n + 1) = 1$.
(a) We've already shown in Exercise (1.1).18 a stronger result that consecutive integers have no common factors, prime or composite, except 1.
For practice, let's do it again.
If $g$ is a common factor of consecutive integers $n$ and $n+1$, then there exist integers $j,k$ such that
$$ n = jg $$
$$ n + 1 = kg $$
This means
$$ j + \frac{1}{g} = k $$
The only way $j,k$ are integers is if $g=1$.
That is, consecutive integers only have 1 as a common factor, and so have no prime factors in common.
(b) We showed in part (a) above that consecutive integers $n, n+1$ only have 1 as a common factor.
The gcd is the greatest common factor, which here is 1.
So $\gcd(n,n+1)=1$.