Is $aZ \cup b Z = \gcd(a, b)Z$? 

I know that $\gcd(a,b)$ is a factor of both $a$ and $b$, but I am not sure how this translates to $aZ \cup b Z = \gcd(a, b)Z$. Therefore I tried to find a counter example to disprove this, but was unsure how to structure the sets $a$ and $b$, then determine a $\gcd$ from that.

Second question, 

Can you prove that $aZ \cap bZ = lcm(a,b)Z$? 

I know that $aZ$ and $bZ \in lcm(a,b)Z$, but I don't know how to prove this equality is equal.