Why is the largest power of prime $p$ that divides $n$ called the "order" of $n$ at $p$?

Question

Largest exponent $a$ of the power $p^a$ of prime $p$ that divides $n$ is called order of $n$ at $p$ and denoted $ord_p (n)$.

Why is this function called order?

Would it be better to call it occurence of $p$ in $n$?

Answer 1

Order functions $o\colon R\rightarrow \mathbb{N}_0$ from a ring $R$ to non-negative integers in general satisfy $o(xy)=o(x)+o(y)$ and $o(x+y)\ge \min\{o(x),o(y)\}$. Since $o(n)=ord_p(n)$ satisfies this properties, it makes sense to call it order function.

Another example: Order functions on the universal enveloping algebra $R=U(\mathfrak{g})$ of a Lie algebra, via Poincare-Birkhoff-Witt basis.