Lately I have been studying the divisibility of some cyclic sums, and I was wondering about the following Conjecture Let it be a set of distinct integers $S=\{x_1,x_2,…,x_n\}$. Then, $$\prod_{n=1}^{n}x_{n}\nmid\left(\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)+\sum_{cyc}\left(\prod_{n=1}^{n-2}x_{n}\right)+…+\sum_{n=1}^nx_n\right)$$ I would appreciate any help regarding this conjecture proof or refutation. I already noted that there exist sets of distinct integers $S=\{x_1,x_2,…,x_n\}$ such that $$\prod_{n=1}^{n}x_{n}\mid\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)$$ […]

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## Divisibility of cyclic sums

- Post author By Q+A Expert
- Post date April 1, 2020
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- Tags ..., $$\prod_{n=1}^{n}x_{n}\nmid\left(\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)+\sum_{cyc}\left(\prod_{n=1}^{n-2}x_{n}\right)+...+\sum_{n=1}^n, $$2*3*6\mid (2*3)+(3*6)+(6*2)$$ Thanks in advance!, 3, 6, and I was wondering about the following Conjecture Let it be a set of distinct integers $S=\{x_1, for $S=\{2, Lately I have been studying the divisibility of some cyclic sums, x_2, x_n\}$ such that $$\prod_{n=1}^{n}x_{n}\mid\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)$$ For example, x_n\}$. Then