On the modular invariance of a prime factor "clock-walk" arithmetic function

I am investigating a novel arithmetic function $f: \mathbb{Z}^+ \to \{0, 1, \dots, 9\}$ that maps an integer to a terminal state on a $\mathbb{Z}/10\mathbb{Z}$ cycle based on its distinct prime signature.

Definition

Let $n \in \mathbb{Z}^+$ have the unique prime factorization $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$. We define the set of distinct prime factors as $\mathcal{P}(n) = \{p_1, p_2, \dots, p_k\}$. The function $f(n)$ is defined as:

$$f(n) = \left( \sum_{p \in \mathcal{P}(n)} p \right) \pmod{10}$$

This can be visualized as a self-intersecting walk on a $10$-point clock diagram starting at $0$, where each unique prime factor dictates a clockwise step size.

Empirical Examples

• $f(14) = (2 + 7) \pmod{10} = 9$
• $f(15) = (3 + 5) \pmod{10} = 8$
• $f(16) = (2) \pmod{10} = 2$
• $f(17) = (17) \pmod{10} = 7$

The Conjecture: Invariance under Multiples of 30

Let $\mathcal{M}_{30}$ be the set of positive multiples of $30$, such that $n \in \mathcal{M}_{30} \implies n = 30m$ for $m \in \mathbb{Z}^+$. Because $30 = 2 \times 3 \times 5$, it is guaranteed that $\{2, 3, 5\} \subseteq \mathcal{P}(n)$. Since $2 + 3 + 5 = 10 \equiv 0 \pmod{10}$, I propose the following invariant property:

Conjecture: For any $n \in \mathcal{M}_{30}$, the value of $f(n)$ is entirely independent of its three smallest prime factors, simplifying to: $$f(30m) = \left( \sum_{p \in \mathcal{P}(30m) \setminus \{2,3,5\}} p \right) \pmod{10}$$

Questions

1. Proof/Counterexample: Does this structural invariance hold universally for all $n \in \mathcal{M}_{30}$, or are there underlying edge cases where this identity breaks down?
2. Generalization: If we extend this to a modulus $M$, under what exact algebraic conditions does a generator set $\mathcal{G} \subset \mathcal{P}(n)$ guarantee a perfect null-orbit reset ($0 \pmod M$) for any multiple of $\prod_{p \in \mathcal{G}} p$?
3. Asymptotic Density: As $n \to \infty$, does the distribution of $f(n)$ converge to a uniform distribution across the residues of $\mathbb{Z}/10\mathbb{Z}$?