I'm having a hard time grasping the concept of finding the number of Sylow p-subgroups. Consider a group $G$ of order $3 \cdot 5 \cdot 7 = 105$. Now, let's find the number of Sylow 3-subgroups for this group: we have
$5 \equiv 2 \pmod 3\\7 \equiv 1 \pmod 3\\35 \equiv 2 \pmod 3$
So $1$ and $7$ are my options, by Sylow's third theorem. How do I determine which, in a none-bruteforce manner? (That is, I consider "we found $2$ so there must be $7$" a bad method. If you disagree strongly about this, feel free to present your argument and convince me, as long as you present a convenient method for the case where it is $1$).
You do need to bruteforce insofar as the answer depends on details of the group. We could have $G\cong (Z_7\rtimes Z_3)\times Z_5$ where there are seven and we could have $G\cong Z_{105}$ where there is just one.