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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?