a) As $a + 2d > a$, there exists some power of a prime $p^k$, that $a + 2d$ is divisible by $p^k$ and $a$ isn't divisible by $p^k$. From the given equality it follows that $a + d$ must be divisible by $p^k$. But then $2(a + d) - (a + 2d) = a$ is divisible by $p^k$, contradicting the choice of $p^k$. This contradiction completes the proof.

b) It's enough to provide an example: $a = 4, d = 2$, then $a + d = 6$ and $a + 4d = 12$. Checking:
$$[a, a + d] = [4, 6] = 12 = [4, 12] = [a, a + 4d].$$