{"id":5433,"date":"2025-12-12T17:26:48","date_gmt":"2025-12-12T21:26:48","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5433"},"modified":"2026-01-08T17:28:29","modified_gmt":"2026-01-08T21:28:29","slug":"number-theory-challenge-12-12-2025","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5433","title":{"rendered":"Number Theory Challenge &#8211; 12\/12\/2025"},"content":{"rendered":"\n<p>Prove that for integers $a$,$b$,$c$, if $9|(a^3+b^3+c^3)$, then at least of them is divisible by $3$.<a href=\"javascript:toggle_visibility('number-theory-chall-12-12-2025')\">\ud83d\udd11<\/a><\/p>\n<div id=\"number-theory-chall-12-12-2025\" style=\"display:none\"><\/p>\n\n\n\n<p><strong>Proof:<\/strong> Let $a=3A+x$, $b=3B+y$, $c=3C+z$, where $A$, $B$, $C$ are integers, $a\\pmod 3=x$, $b\\pmod 3=y$, $c\\pmod 3=z$. Therefore $0\\le x,y,z\\le 2$. <\/p>\n\n\n\n<p>Without loss of generality, we assume that $0\\le x\\le y\\le z\\le 2$. We have<\/p>\n\n\n\n<p>$$(a^3+b^3+c^3)\\pmod 9=((3A+x)^3+(3B+y)^3+(3C+z)^3)\\pmod 9$$ $$\\ \\ \\ =(x^3+y^3+z^3)\\pmod 9$$<\/p>\n\n\n\n<p>If $9|(a^3+b^3+c^3)$, then $9|(x^3+y^3+z^3)$. If $x\\ne 0$, then none of $x$,$y$,$z$ is $0$, and none of $a$,$b$,$c$ is divisible by $3$, the $(x,y,z)$ values can only be $(1,1,1)$, $(1,1,2)$, $(1,2,2)$, $(2,2,2)$, which result in $(x^3+y^3+z^3)$ values as $3$, $10$, $17$, $24$, with none of them divisible by $9$. Therefore, $x=0$, and at least one of $a$, $b$, $c$ is divisible by $3$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Prove that for integers $a$,$b$,$c$, if $9|(a^3+b^3+c^3)$, then at least of them is divisible by $3$.\ud83d\udd11 Proof: Let $a=3A+x$, $b=3B+y$, $c=3C+z$, where $A$, $B$, $C$ are integers, $a\\pmod 3=x$, $b\\pmod 3=y$, $c\\pmod 3=z$. Therefore $0\\le x,y,z\\le 2$. Without loss of &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=5433\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[16],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5433"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5433"}],"version-history":[{"count":16,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5433\/revisions"}],"predecessor-version":[{"id":5449,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5433\/revisions\/5449"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}