Subgroups of $\mathbb{Z}$? I have this set:
$$I_a = \{ax+(7-a^2)y : x, y \in \mathbb{Z}\}$$
with $a$ integer. I have two tasks over it: 
1) Prove that $I_a$ is subgroup of  $\mathbb{Z}$;
2) Fully characterize this subgroup. 
I did the first task, but for the second I was thinking on cases, that is, if $a$ is equal to $7k,  7k\pm 1,7k\pm 2$ or  $7k \pm 3$, for some $k \in \mathbb{Z}$. I was successful with $a= 7k$ (this is: $\langle 7 \rangle = I_a$), but I can't with the other cases. Some hints?  

First, can you show that 7 is in $I_a$? 
Then, can you show that if there is even one number in $I_a$ that isn't a multiple of 7, then $I_a$ is all of $\bf Z$? 
