{"id":3847,"date":"2023-09-08T13:55:00","date_gmt":"2023-09-08T17:55:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=3847"},"modified":"2024-10-25T09:32:26","modified_gmt":"2024-10-25T13:32:26","slug":"mathcounts-training-number-theory-4","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=3847","title":{"rendered":"MathCounts Training &#8211; Number Theory &#8211; 4"},"content":{"rendered":"\n<ol><li>________ What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?$$ $$<\/li><li>________ A digit can be placed in each of the boxes for hundreds and units digits to form the least positive five-digit number divisible by 96. What is the ratio between the smaller digit to the larger digit? Express your answer as a common fraction. $$21\\boxed{\\phantom{8} }7\\boxed{\\phantom{8}}$$<\/li><li>________ In order, the first four terms of a sequence are 2, 6, 12 and 72, where each team, beginning with the third term, is the product of the two proceeding terms. If the ninth term is $2^a3^b$, what is the value of $a+b$? $$ $$<\/li><li>________ What is the sum of the numbers less than 200 that have exactly 9 divisors? $$ $$<\/li><li>________ Given that $A=\\{\\dfrac{n}{24}\\}$, $n$ is a natural number, $gcd(n,24)=1$, and $\\dfrac{n}{24}\\lt 2$, what is the sum of the elements in $A$? $$ $$<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"<p>________ What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?$$ $$ ________ A digit can be placed in each of the boxes for hundreds &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=3847\">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":[18,11,16],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3847"}],"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=3847"}],"version-history":[{"count":9,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3847\/revisions"}],"predecessor-version":[{"id":3856,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/3847\/revisions\/3856"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3847"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3847"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3847"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}