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We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\\le d\\le 36$), the smallest prime $p\\equiv c\\pmod d$ with $p\\ge(2dn-c)/(d-1)$ is the least positive integer $m$ with $2r(d)k(dk-c)\\ (k=1,\\ldots,n)$ pairwise distinct modulo $m$, where $r(d)$ is the radical of $d$. 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