If we consider the integers modulo a prime $p$, then for every $x \not \equiv 0$ (mod $p$), we can get any $b \not \equiv 0$ by adding $x$ a number of times to itself.
Is the same true for multiplication? That is, is there a number $x$ which, multiplied the right number of times to itself will give any other number modulo $p$?
I have done some research and it seams that, if this is true, it makes modular integers a cyclic group over multiplication. From what I found, it might be related to Lagrange's theorem. However, I would be interested in a lower level answer, if one exists. I also believe Fermat's little theorem would follow from this (though there are more straightforward ways to get there).
This write-up by Keith Conrad contains seven proofs of this result. More generally, any finite subgroup of the multiplicative group of a field is cyclic.