It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least $n_p(p^n-1)$ elements, where $n_p$ is the number of Sylow $p$-subgroups.
It is explained that the reason this is the case is because distinct Sylow $p$-subgroups intersect only at the identity, which somehow follows from Lagrange's Theorem.
I cannot see why this is true.
Can anyone quicker than I tell me why? I know it's probably very obvious.
Note: This isn't a homework question, so if the answer is obvious I'd really just appreciate knowing why.