Indivisible

Use the Inclusion-Exclusion formula: (see https://brilliant.org/wiki/principle-of-inclusion-and-exclusion-pie/ )

Notation: let $[n] =$ {integers from 1 to n}, and, for our purposes, $[k,n] =$ {multiples of k less than or equal to n}

We are looking for $|[1000] \setminus \{[3,1000] \cup [5,1000] \cup [7,1000] \}|$

Let $X=[1000]$ , and $X_{t} = [t,1000]$ for $t \in \{3,5,7,15,21,35,105\}$

Let $X^*= X \setminus (X_3 \cup X_5 \cup X_7)$

Note : for $p,q \in \{3,5,7\}$ , $X_p \cap X_q = X_{pq}$ . Similarly, $X_3 \cap X_5 \cap X_7 = X_{105}$

By I-E, $|X^*| = |X| - (|X_3| + |X_5| + |X_7|) + (|X_{15}| + |X_{21}| + |X_{35}|) - |X_{105}|$

Note also $|X_k| = \lfloor({\frac{1000}{k}})\rfloor$

Thus $|X^*| = 1000 - (333 + 200 + 142 ) + (66 + 47 + 28) -9$ , i.e. $|X^*| = 457$ .