Relationship between elements of a group

Question

Let $G$ be a group and $a,b \in G$. If $a^{17} = b^{17}$ and $a^{30} = b^{30}$ then

1. $a = b$
2. $ab=ba$ and $o(a) \neq o(b)$
3. $a = b^{-1}$ and $o(a) \neq o(b)$
4. $o(a) = o(b)$ and $a \neq b$

Since $\gcd(17,30)=1$ therefore $a$ should be equal to $b$ but I'm not really sure.

Answer 1

There are integers $x,y$ such that $17x+30y=1$, so that $$ a=a^1=a^{17x+30y}=b^{17x+30y}=b^1=b. $$