There is no submitter.<\/p>\n
Hint:
\nFor 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 $latex a$, $latex b$.<\/p>\n
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 $latex v_{p_i}(x_i) + v_{p_i}(x_{i+1}) \\lneq v_{p_i}(n)$.<\/p>\n
Observe the primes $latex p_1$, $latex p_2$, $latex \\ldots$, $latex p_9$.<\/p>\n","protected":false},"excerpt":{"rendered":"
There is no submitter. 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… <\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"_links":{"self":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts\/1993"}],"collection":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1993"}],"version-history":[{"count":3,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts\/1993\/revisions"}],"predecessor-version":[{"id":1996,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts\/1993\/revisions\/1996"}],"wp:attachment":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1993"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1993"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1993"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}