Currently I am reading Simpson's Subsystems of Second Order Arithmetic, Chapter 2, Recursive Comprehension (RCA$_0$)
The author stated the following theorem at page $67.$
Theorem $2.5$ The following is provable in RCA$_0$. For any finite set $X\subseteq \mathbb{N},$ there exist $k,m$ and $n\in \mathbb{N}$ such that $$\forall i(i\in X\leftrightarrow(i<k \wedge m(i+1)+1 \text{ divides }n)).$$
I think $m$ is an upper bound of $X$ and $m$ is the lowest common multiple of all elements of $X.$
However, I am not able to figure out the role of $n.$
The author also mentions that each finite set of natural numbers has a unique code.