First i calculated how many numbers are divisible by 2 and 5
$\frac{10000}{2}+\frac{10000}{5}-\frac{10000}{2*5}=6000$
Then i calculated how many numbers are not divisible by 3 or 4
$10000-(\frac{10000}{3}+\frac{10000}{4}-\frac{10000}{3*4})=5000 $
Now i do not know if these steps are correct or even necessary, but i do not know how to get the final result.
Thanks!
Best Answer
If they are divisible by $2$ and $5$ ,then these numbers can be divided by $10$. So how many numbers are there $[1,10000]$ divisible by $10$ . ?
Answer can be found by Gauss formula such that $\frac{10000-10}{10} + 1 =1000$.
Now assume that our universal set is consisted of these $1000$ numbers . Then , we should eliminate those which are not divisible by $3$ or $4$.
Divisible by $10$ but not $3$ : Divisible by $10$ - (divisible $3$ and $10$) = Divisible by $10$ divisible by $30$ = $1000 - \frac{9990 - 30}{30} + 1 =1000-333=667$
Divisible by $10$ but not $4$ : Divisible by $10$ - (divisible $4$ and $10$) = Divisible by $10$ divisible by $20$ = $1000 - \frac{10000 - 20}{20} + 1 =1000-499=501$
Divisible by $10$ but not $4$ and $3$ at the same time : Divisible by $10$ - (divisible $12$ and $10$) = Divisible by $10$ divisible by $60$ = $1000 - \frac{9960 - 60}{60} + 1 =1000-166=834$
Then by inclusion exlusion = $667+501-834 =334$
Hence = $1000-334 =666$