Let $p\in \mathbb{Z}_{\geq 1}$ be a prime number, and consider the ring $\mathbb{Z}_{p}$ of $p$-adic integers. Now every rational number $n \in \mathbb{Q}$ is a $p$-adic integer for all but finitely many primes $p$, so for any given $n \in \mathbb{Q}$ let us ignore the finitely many primes where this is not the case.
Every $n \in \mathbb{Z}_{p}$ has a unique expression of the form $$a_0+a_1p+a_2p^2+\dots$$ where $0 \leq a_i \leq p-1$ for every $i\geq 0$.
Now for any given $n \in \mathbb{Q}$ there are only finitely many primes $p$ such that $n \in \mathbb{Z}_{p}$ and $a_0=0$. In other words, $n$ is a unit element in $\mathbb{Z}_{p}$ for almost all prime $p$.
My question is: can we say the same about the other coefficients $a_1,a_2,\dots$? For a given $n \in \mathbb{Q}$, is $a_1=0$ for only finitely many primes $p$?
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As for context, this problem is a toy instance of a subproblem that arose as part of my attempt at this question I had asked earlier: When can an algebraic number be approximated by a $p$-adic number? Evidently, that question is not juicy enough to attract eyeballs, and requires more background. However, this simpler formulation captures the core of that problem (I think). I apologize if the solution turns out to be trivial, but clearly I am missing the picture.
UPDATE: As was pointed out in the comments (sorry and thanks!), I had worded the question wrong earlier. I think I have rectified it now, but if its still silly, I'll edit it again.