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 2 of 5: Fix a prime power exactly dividing n

Previous step Next step

q=p^e \parallel n \implies p \mid a_1 \text{ or } p \mid (a_2-1)

Detailed analysis

Fix a prime $p$ and exponent $e\ge1$ such that $q=p^e$ exactly divides $n$ (written $q\parallel n$ ). Since $q$ divides $a_1(a_2-1)$ , the prime $p$ divides $a_1$ or divides $a_2-1$ . We examine each case and show it forces $a_i \pmod q$ to be the same constant for every $i$ .