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 3 of 5: Case p | a₁: chase the cycle backward

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p \mid a_1 \implies q \mid a_i \text{ for every } i=1,\dots,k

Detailed analysis

If $p\mid a_1$ then $p\nmid(a_1-1)$ , so $a_1-1$ is invertible modulo $q$ . From $a_k(a_1-1)\equiv0\pmod q$ we get $q\mid a_k$ . Now $p\mid a_k$ makes $a_k-1$ invertible mod $q$ , so $a_{k-1}(a_k-1)\equiv0\pmod q$ gives $q\mid a_{k-1}$ . Repeating this backward through the relations $i=k-1,k-2,\dots,1$ shows $q\mid a_i$ for every $i=1,\dots,k$ : all the $a_i$ are $\equiv 0\pmod q$ .