Finding infinitely many Myhill–Nerode equivalence classes
Let's consider your second example (the first example is easier). We will be interested in
$$
F(m) = \{ n : 0^m 1^n \in L \} = \{ n : \gcd(m,n) = 1 \}.
$$
If $F(m_1) \neq F(m_2)$, then $m_1$ and $m_2$ belong to two different equivalence classes.
From $F(m)$ you can recover the set of primes dividing $m$: $p \in F(m)$ iff $p \not\mid m$. In particular, this shows that if $p_1 \neq p_2$ are two different primes, then $F(p_1) \neq F(p_2)$. Since there are infinitely many primes, we conclude that there are infinitely many different equivalence classes.
The first example can be analyzed in a similar way (considering only odd $m$).