Given a positive integer $x$,how many possible remainders can you get after dividing $x$ by positive integers smaller than it?
I have been thinking about this question for some time.Here is an explicit example.
Let the number be $100$.No matter what number you choose that is less than $100$,you can never get a remainder of $50$.So my question is,how many possible remainders are there after division of a number $x$ [such as $100$] by other numbers?This should require some modular arithmetic,since it deals with remainders.
Now obviously,all numbers greater than $\dfrac{x}{2}$ generate remainders that are integers smaller than $\dfrac{x}{2}$.Does it mean that when the divisor is smaller than x,the only remainders not possible are $x/2$ and integers greater than $x/2$?
Some form of hint will be appreciated.
Happy Yuletime,everyone.