Problem on Principle of Inclusion-Exclusion How many integers $1, 2,....., 11000$ are invertible modulo $880$?
$880$ can be rewritten as $2^4\cdot5\cdot11$.
So I am supposed to find the number of integers in this range that have $2$, $5$ or $11$ as a divisor and then subtract that value from $11000$. 
So If I divide $11000$ by each of $2$, $5$ and $11$, I get cardinalities of $5500$, $2200$, and $1000$ respectively. But how exactly am I supposed to find how many integers there are that have both $2$ and $5$ as a divisor, $2$ and $11$ as a divisor, and $5$ and $11$ as a divisor? How am I supposed to find the amount of integers that have all three numbers as a divisor? 
Any help?

Denote the set of integers which are divisible by $n$ in the range $\{1,2,\cdots,11000\}$ by $N(n)$. We wish to find $|N(2)\cup N(5)\cup N(11)|$. We can use inclusion exclusion.
$$|N(2)\cup N(5)\cup N(11)|=|N(2)|+|N(5)|+|N(11)|-|N(2)\cap N(5)|-|N(2)\cap N(11)|-|N(5)\cap N(11)|+|N(2)\cap N(5)\cap N(11)|$$
As stated in the comments, to find the size of the intersections, you need to find out how many integers are divisible by multiple numbers. If a number is divisible by $x$ and is divisible by $y$, and $x$ and $y$ are coprime (as they are in this example, $2$,$5$,$11$), then the number is divisible by $xy$. Use this to find the size of the set.
