62nd Ukrainian National Mathematical Olympiad, Third Round, Second Tour

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: $$[a,a+d]=[4,6]=12=[4,12]=[a,a+4d].$$