So I'm taking an introductory number theory course as an undergraduate, and this particular "genre" of questions really just has me stumped. Pick a prime [tex]p[/tex] such that [tex]p[/tex] is odd. Now, take various sums up of natural numbers from 1 to [tex]p[/tex], and show that the results are divisible by [tex]p[/tex]. For example, consider, [tex]n = 1\cdot 2 + 2\cdot 3 + \cdots + (p-1)\cdot p[/tex] And show that it is divisible by [tex]p[/tex]. Now in my notes, I have something about polynomial congruences, and how the coefficients must all be equal. And I have the congruence, [tex]x^p - x \equiv (x-1)(x-2)\cdots (x-p) \bmod p[/tex] Which apparently helped find the result, but I'm not sure why. It made sense when I wrote it down, but now I'm drawing a blank. Can anyone maybe help me fill in the gaps?