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Let $n$ be a positive integer and let $a_1, . . . , a_k (k \geq 2)$ be distinct integers in the set ${{{1, . . . ,n}}}$ such that $n$ divides $a_i(a_{i+1} −1)$ for $i = 1, . . . ,k−1$. Prove that $n$ does not divide $a_k(a_1−1)$.
[easiest IMO (contest) problem of 21st century................to me ]

If anybody need a hint...