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For question (a): Each number n in the sum is lower than p. So, (n,p)=1.
Hence, we can apply Fermat's Little Theorem:
Let's multiply both sides by n.
Isn't it ?
Last edited by Klaus; June 23rd 2008 at 06:57 AM.
For (a) we can strengthen the problem. if and if .
For (b) note . Let .
Let be remainder of mod .
Thus, thus and .
By Wilson's theorem it follows that .
Also because divides .
If we let it satisfies both conditions.
For the second problem let be the order of mod . Let . Write where . Then, . Since is the order of it means if . But since and it follows that . And thus .