"How many different integers does this give us?" How many unique integers can you get from
$\lceil2012/n\rceil$
where $n$ is a positive integer?
I don't know at all where to begin to approach this problem. I thought it maybe had something to do with which factors the number has or anything like that, but I couldn't find any obvious pattern.
 A: Here is a program that will print 89 like @joriki said!!
class Program
    {

        static void Main(string[] args)
        {
            int n = 1,c=0;
            double prev = 0;
            for (n = 1 ; n <= 2012; n++)
            {
                double curr=Math.Ceiling((double)2012/n);
                if (prev != curr) c++;
                //Console.WriteLine(curr.ToString());
                prev=curr;
            }
            Console.WriteLine(c.ToString());
            Console.Read();
        }
    }

A: If the difference
$$
\frac{2012}n-\frac{2012}{n+1}=\frac{2012}{n(n+1)}
$$
is less than $1$, then there cannot be an integer between $\lceil2012/(n+1)\rceil$ and $\lceil2012/n\rceil$; whereas if the difference is greater than $1$, then $\lceil2012/(n+1)\rceil$ and $\lceil2012/n\rceil$ cannot be equal. The crossover happens between $n=44$ and $n=45$. Thus the $44$ values for $n=1$ to $n=44$ are all distinct, and the integers $n\ge45$ exactly cover the $45$ values from $1$ to $\lceil2012/45\rceil=45$, for a total of $44+45=89$ different values.
