The question has 3 parts:

My Attempt:

(a)Since the cyclic group generated by a is $\displaystyle \left\langle a \right\rangle = \{ za | z \in \mathbb{Z} \}$, I think it would follow that for distinct $\displaystyle a_1,a_2,...,a_n \in \mathbb{Z}$

$\displaystyle \left\langle a_1,...,a_n \right\rangle = \{ z_1a_1 +...+z_na_n\}$

but I don't know how to actually prove this. I appreciate any hints to get started.

(b)Though I'm not sure if it helps... but I think in general $\displaystyle \left\langle ma \right\rangle \cap \left\langle na \right\rangle = \left\langle ka \right\rangle$ where $\displaystyle k =lcm(m,n) \mod\ \mathbb{Z}$. Is that correct?

(c)Is the gcd(a,b) required to be 1 (so that a and b are relatively prime)? If so I get

$\displaystyle \left\langle a,b \right\rangle = \{ au+bv | u,v \in \mathbb{Z}\}= \left\langle h \right\rangle$

$\displaystyle \implies au+bv=1$

Is "h" equal to 1?