Question

Let $S$ be the set of integers which are divisible by 5 , and let $T$ be the set of integer which are divisible by 7. Find the number of positive integers less than 1000 and not in ( $S\cup T$ ).

Answer 1

Answer: 686

$n(S)=5+10+\dots \dots +1000=200-1=199$
$n(T)=7+14+\dots \dots +994=142$
$n(S\cap T)=35+70+\dots \dots +980=28$
$n(S\cup T)=341-28=313=n(S)+n(T)-n(S\cap T)$

$\text{ Total }=999$
$n(\overline{S}\cap \overline{T})=999-313=686$