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Hint:
For any positive integer $latex m$ and prime $latex p$, let $latex v_p(m)$ be the exponent of $latex p$ in the prime factorization of $latex m$. Then $latex v(ab) = v_p(a) + v_p(b)$ for any positive integers $latex a$, $latex b$.

Assume that $latex n$ satisfies the condition above with the numbers on the circle being x1, $latex x_2$, $latex \ldots$, $latex x_9$. Set $latex x_{10} = x_1$ and $latex x_{11} = x_2$ (to simplify the notation). For each $latex 1 \le i \le 9$, since $latex n$ does not divide $latex x_ix_{i+1}$, there must be a prime $latex p$ such that $latex v_p(x_i x_{i+1}) \lneq v_p(n)$. Let $latex p_i$ be one such prime, so $latex v_{p_i}(x_i) + v_{p_i}(x_{i+1}) \lneq v_{p_i}(n)$.

Observe the primes $latex p_1$, $latex p_2$, $latex \ldots$, $latex p_9$.