Originally Posted by

**yasste** Hey you all,

I hope you have better mathematical skills than me. I only study math at a low level since i am study to become a teacher. Sadly, this leaves me with only having a basic knowledge and this is probably why I am stuck with this both problems I have been given to solve and my pass and grade depens on it. But i am stuck and becoming desperate, so i hope there is someone who can help me solving it. I don't necessary need to know the whole solution. But maybe some tips and idications how i will solve it. Thank you all so much.

(by the way I hope i use the right mathematical language, cos i am only used to the german words in the mathematical sense)

So these are my problems:

1) Determine the rest of

$\displaystyle \sum\limits_{i=1}^{n} 10^{10^i}$ when divided by 7.

Using basic properties of powers and Fermat's Little Theorem, it's easy to prove that $\displaystyle 3^{10^i}=4\!\!\!\pmod 7\,\,\,\forall\,i\in\mathbb{N}$, so this sum is just $\displaystyle 4n\!\!\!\mod 7$ ...

2) Prove that

$\displaystyle n^{4k+1} \equiv n\, mod\, m, $ for all n, k $\displaystyle \in \mathbb{N} \iff m|30 $