Here, we can apply the property of set. Let $D_n$ denote divisibility by $n,$ $D_{n_1,n_2}$ denote divisibility by both $n_1$ and $n_2$ and so on.
$N(D_3 \cup D_5 \cup D_7)=N(D_3)+N(D_5)+N(D_7) -N(D_{3,5})-N(D_{ 5,7})-N(D_{3,7})+N(D_{3,5,7})$
$\quad \quad =166+100+71-33-14-23+4$
$\quad \quad =271$