Summary: It is shown that the set of periods of any additive cellular automata \(F\), where the addition is done modulo a prime \(p\), can be determined using some simple conditions on the coefficients in the linear expression of \(F\). In particular, we establish that the set of periods has only four possibilities: \(\{1,m\}\) for some \(m\) where \(1\leq m<p\), \(\mathbb{N} \setminus\{p^m:m\in\mathbb{N}\}\), \(\mathbb{N}\setminus\{2 p^m:m \in\mathbb{N}\cup \{0\}\}\) or the whole set \(\mathbb{N}=\{1,2,3,\dots\}\). Using our results, the set of periods of any additive cellular automata, where the addition is done modulo a square-free positive integer, is easily obtained.