[ Divisibility of numbers. General properties ] $[$ Classical combinatorics (other) $]$

How many natural numbers less than a thousand are not divisible by either 5 or 7?
First, we will strike out from the set of numbers $1,2, \ldots, 999$ the numbers that are multiples of 5; their quantity is $[999 / 5]=199$. Then, from the same set of numbers $1,2, \ldots, 999$, we will strike out the numbers that are multiples of 7; their quantity is $[999 / 7]=142$. In this process, numbers that are multiples of 35 will be struck out twice. Their quantity is [999/35] $=28$. Therefore, in total, we have struck out $199+142-28=313$ numbers, and $999-313=686$ numbers remain.

## Answer

686 numbers.