By Sylow's Theorem, the number of Sylow $p$-subgroups, $n_p$, must divide $qr$ and be congruent to $1$ modulo $p$. Similarly, the number of Sylow $q$-subgroups, $n_q$, must divide $pr$ and be congruent to $1$ modulo $q$. Finally, the number of Sylow
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