If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$?

Question

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $$N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$$

Answer 1

Notice that $\gcd\left(\left|N_i\right|,\left|N_j\right|\right)=1$ if $i\neq j$

Hence, $$\gcd\left(\left|N_i\right|,\left|\prod_{k\neq i} N_k\right|\right)=1$$

Thus, their intersection is trivial.