How do I accurately count the integers(1-1000) that are not divisible by 3,4,5,6? I have the general algorithm here that my teacher gave us( see full at http://i.imgur.com/pbzQb.png)   )
To count we just divide, correct?
like - 1000/3 = 333 ?
What is the sigma notation used here(it's for principle of inclusion, exclusion I thiink)? Also, I am wondering about  the boundary cases: near 1000.
Update: I am close I think i can do this :)

$\sum_{i=1}^4 N_i$ is shorthand for $N_1+N_2+N_3+N_4$. 
Similarly, $\sum_{i,j=1}^4N_{ij}$ means write down all the numbers of the form $N_{ij}$ where $i$ and $j$ take on the values from 1 to 4, and add them up. However, that's not really what you want: you really want $\sum_{1\le i\lt j\le4}N_{ij}$, which is $N_{12}+N_{13}+N_{14}+N_{23}+N_{24}+N_{34}$. 
EDIT: Actually, it appears from your $N_{3,4,5,6}$ that what you really want is for your sums to go from 3 to 6, not from 1 to 4, e.g., you want $\sum_{i=3}^6N_i=N_3+N_4+N_5+N_6$. 
