Problem 1

Let

n, k \ge 2

be positive integers and let

a_1, a_2, \dots, a_k

be distinct integers in the set

\{1, 2, \dots, n\}

such that

n

divides

a_i(a_{i+1} - 1)

for

i = 1, 2, \dots, k - 1

. Prove that

n

does not divide

a_k(a_1 - 1)

Step 4 of 5: Otherwise p | (a₂−1): chase the cycle forward

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p \mid (a_2-1) \implies a_i \equiv 1 \pmod q \text{ for every } i=1,\dots,k

Detailed analysis

Otherwise $p\nmid a_1$ , so $a_1$ is invertible mod $q$ and $a_1(a_2-1)\equiv0\pmod q$ forces $a_2\equiv1\pmod q$ . Then $a_2$ is invertible mod $q$ , so $a_2(a_3-1)\equiv0\pmod q$ gives $a_3\equiv1\pmod q$ . Continuing forward through $i=2,3,\dots,k-1$ gives $a_i\equiv1\pmod q$ for $i=2,\dots,k$ ; finally the assumed relation $a_k(a_1-1)\equiv0\pmod q$ (with $a_k$ invertible) forces $a_1\equiv1\pmod q$ as well. So all the $a_i$ are $\equiv 1\pmod q$ .