$\newcommand\N{\mathbb N} \newcommand\ceil[1]{\lceil#1\rceil}$Let $a_1,\dots, a_k\in \N$ be an arbitrary finite set of positive integers. Can we find a prime number $p$ such that $p>k$ (preferably $p\gg k$) and a natural number $n\in\N$ such that $$a_1n,a_2n,\dots,a_kn$$ are equivalent to integers in $[p/k,p-p/k]$ mod $p$?
Note: $[a,b]$ here just means a closed interval in the real line.
Not always. Indeed, let $a_i=i$ for each $i$ and $n$ be any natural number distinct from $1$ and $p$. Since there are $p$ residues modulo $p$ and $k+1$ numbers of the form $ni$ for $i=0,\dots, k$, by pigeonhole principle, there exist $0\le i<j\le k$ and an integer $\Delta$ with $|\Delta|\le\frac p{k+1} $ such that $jn\equiv in+\Delta\pmod p$. Then $0<j-i\le k$ and $(j-i)n$ is equivalent mod $p$ to a unique integer in $[0,p-1]$ which is $|\Delta|\le\frac p{k+1}<\frac p{k}$ or $p-|\Delta|\ge p-\frac p{k+1}>p-\frac p{k}$.