Prove that (relatively prime means their gcd is 1):
(a) two consecutive integers are relatively prime.
(b) two consecutive odd integers are relatively prime.
(a) Consider two consecutive integers $m$ and $n=m+1$.
If $a \mid m$, then $m = ak$, for some integer $k$.
If we also have $a \mid n$, then $m+1= aj$, for some integer $j$.
Combining, we have
$$aj = ak + 1$$
Since $a \neq 0$, we can divide through,
$$j = k + \frac{1}{a}$$
The only way $j$ and $k$ are integers is if $a=1$.
That is, the common divisor of two consecutive integers is 1. Which is the same as concluding they are relatively prime.
(b) Consider two consecutive odd integers $m = 2k+1$ and $n=2k+3$, for some integer $k$.
If $a$ is a common divisor, then we have, for some integers $j, k$,
$$2k + 1 = ak$$
$$2k + 3 = aj$$
Subtracting, and re-arranging, we have
$$j= k + \frac{2}{a}$$
For $j$ and $k$ to be integers, $a$ must be 1 or 2. However, since $m$ and $n$ are odd, they don't have 2 as a divisor, so only $a=1$ remains as a possibility.
That is, the common divisor of two consecutive odd numbers is 1. Which is the same as concluding they are relatively prime.