I'm searching for a reference for the following result, so as to avoid writing a full proof in a paper. Alternatively, if a one-liner exists, I'd be glad to know it!
Theorem: Let $a, b$ be two positive integers. Then there is a finite set $N$ of positive integers smaller than $\mbox{lcm}(a, b)$ such that: $$\{k_1a + k_2b \mid k_1, k_2 \in \mathbb{N}\} = \{\mbox{lcm}(a, b) + k\times\gcd(a, b) \mid k \in \mathbb{N}\} \cup N.$$
I'm also interested in its generalization to any number of integers: Say a set of integers is a linear set with $n$ periods if it can be written as: $$\{c_0 + \sum_{i=1}^n k_i \times c_i \mid k_i \in \mathbb{N}\}.$$ Then:
Theorem: Any linear set is the union of a finite set and a linear set with one period.
Thanks!