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The following is a Combinatorial proof of Fermat's Little Theorem from Arthur Engel's book:Problem Solving Strategies.

We have pearls with $a$ colors. From these we make necklaces with exactly $p$ pearls. First, we make a string of pearls. There are $a^p$ different strings. If we throw away the a one-colored strings $a^p − a$ strings will remain. We connect the ends of each string to get necklaces. We find that two strings that differ only by a cyclic permutation of its pearls result in indistinguishable necklaces. But there are $p$ cyclic permutations of p pearls on a string. Hence the number of distinct necklaces is $(a^p − a)/p$. Because of its interpretation this is an integer. So $$p | (a^p − a)$$.

I can't figure out how the primality of $p$ is utilised in this proof. Please Explain.

Joffan
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  • There are $p$ cyclic permutations of $p$ pearls on a string, where not all the pearls are the same, precisely because $p$ is prime. If $p$ is not prime you can arrange the pearls in a periodic pattern whose period divides $p$, and then you only have as many distinct cyclic permutations of the pearls as the length of the period. – Vik78 Jan 12 '17 at 09:04

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The primality of $p$ guarantees that there is no shorter-period repeating pattern possible. For non-prime necklace lengths, you may have (for example) three blocks of a repeating pattern, in which case a different number of strings are equivalent when made into necklaces. Because $p$ is prime, you cannot divide it into some number of equal blocks.

Joffan
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