For any integers $a,b,n$ in $\mathbb{Z}$ such that $n\mid a$ and $n\mid b$, which of the following must be true?

Step 1 (Apply Divisibility Definition): By definition of divisibility, if $n\mid a$ and $n\mid b$, there exist integers $k_1$ and $k_2$ such that:

Step 2 (Evaluate the Sum): Add these two equations together:

Since the set of integers is closed under addition, $k_1+k_2$ is also an integer. By the definition of divisibility, this proves that $n$ divides $a+b$:

Step 3 (Conclusion): The correct option is n | (a+b), which is at option index 0.