Marinada '24

[lang=hr] Nakon što je upoznao brucošice, primijetio je da ih je $p$ za neki prost broj, pa je onima koje nije eliminirao u prethodnom zadatku dodijelio nove brojeve $1,2,\dots p$. Naravno, drugi deal-breaker mu je ako za curu s (novom) oznakom $a$ postoji neki $1 \le x \le p-2$ takav da $p \mid a^x-1$. Koliko je brucošica ostalo nakon ovog eliminacijskog testa? [/lang] [lang=en] After meeting the freshmen girls, he noticed that there were $p$ of them for some prime number, so he assigned new numbers $1, 2, \dots, p$ to those who hadn't been eliminated in the previous task. Naturally, his second deal-breaker is if, for a girl with (new) label $a$, there exists some $1 \le x \le p-2$ such that $p \mid a^x - 1$. How many freshmen girls remain after this elimination test? [/lang]