Trivial intersections of Sylow subgroups strengthen the Sylow congruence ($n_p\equiv 1(\mathrm{mod}|P|)$)

Suppose distinct Sylow subgroups for $p$ in a finite group have trivial intersections. Conjugation by one of them, $P$, fixes $P$. The stabilizer of any other $Q$ is $H=P\cap N_G(Q)$. Since $H$ normalizes $Q$, the product $HQ$ is a subgroup of $p$-power order; maximality of $Q$ gives $H\le Q$. Thus $H\le P\cap Q=1$. All other orbits have size $|P|$, proving the congruence.