The method I've been using to prove the closure of a set $(\Bbb Z/N\Bbb Z, +)$ under addition is to list the pairs of possible elements, apply the operation and determine if the result has the same form as the elements of this set. For example:
$Nk + Nk$
$Nk+Nk+1$
$Nk+Nk+2$
$Nk+Nk+3$
$...$
$Nk+1+Nk+1$
$Nk+1+Nk+2$
$Nk+1+Nk+3$
$...$
$Nk+N-1+Nk+N-1$
This is easy for $\Bbb Z/2Z$ and $\Bbb Z/3Z$ but becomes tedious for $\Bbb Z/5\Bbb Z$ and so on.
What is a better way to test if a set like $(\Bbb Z/N\Bbb Z,+)$ is closed?
The set ${\Bbb Z}/N{\Bbb Z}$ is the set of cosets $a+N{\Bbb Z}$, which they $$a+N{\Bbb Z}=\{a+Nk:k\in\Bbb Z\}.$$
The operation among them is $$(a+N{\Bbb Z})+(b+N{\Bbb Z})=a+b+N{\Bbb Z}.$$
For an elementary grasp think as: $$0+5{\Bbb Z}=\{\mbox{multiples of $5$}\}$$ $$1+5{\Bbb Z}=\{\mbox{multiples of $5$ plus $1$}\}$$ $$2+5{\Bbb Z}=\{\mbox{multiples of $5$ plus $2$}\}$$ $$3+5{\Bbb Z}=\{\mbox{multiples of $5$ plus $3$}\}$$ $$4+5{\Bbb Z}=\{\mbox{multiples of $5$ plus $4$}\}$$
Note that $5+5{\Bbb Z}=\{\mbox{multiples of $5$ plus $5$}\}=0+5{\Bbb Z}$, but also $6+5{\Bbb Z}=1+5{\Bbb Z}$, etc.