up vote 1 down vote First, it will be easier to determine the set $$I = \{3k+5j | k,j \in \mathbb{Z}\}.$$ This set is an ideal over $\mathbb{Z}$. That is, if $a,b \in I$ then $a+b \in I$ ($I+I \subset I$), and if $a \in I$ and $z \in \mathbb{Z}$, then