Task A-1.7. (10 points)

How many natural numbers less than 2011 are divisible by at least one of the numbers 2 and 7, but are not divisible by the number 5?
## First solution.

The number of numbers less than 2011 that are divisible by 2 is $\left\lfloor\frac{2010}{2}\right\rfloor=1005$

Among them, $\left\lfloor\frac{1005}{5}\right\rfloor=201$ numbers are divisible by 5

so there are $1005-201=804$ numbers less than 2011 divisible by 2 that are not divisible by 5.

Similarly, with 7, $\left\lfloor\frac{2010}{7}\right\rfloor=287$ numbers less than 2011 are divisible by 7,

and among them, $\left\lfloor\frac{287}{5}\right\rfloor=57$ numbers are divisible by 5

so there are $287-57=230$ numbers divisible by 7 that are not divisible by 5.

With 14, $\left\lfloor\frac{2010}{14}\right\rfloor=143$ numbers are divisible,

of which $\left\lfloor\frac{143}{5}\right\rfloor=28$ are divisible by 5

so there are $143-28=115$ numbers divisible by 14 but not by 5.

Finally, the desired number is $804+230-115=919$.