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?  

To ellaborate on my comment:
Hint: $I_a=\langle a\rangle+\langle 7-a^2\rangle=\langle \gcd(a,7-a^2)\rangle$ 
(where $\gcd$ is the greatest common divisor)
Denote $d=\gcd(a,7-a^2)$. $d$ divides $a$, and $d$ divides $7-a^2$ (clearly $d$ divides $a^2$). Can you proceed? 
