{"id":2298,"date":"2023-11-08T03:30:00","date_gmt":"2023-11-08T07:30:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2298"},"modified":"2024-10-25T12:12:44","modified_gmt":"2024-10-25T16:12:44","slug":"number-theory-challenge-2","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2298","title":{"rendered":"Number Theory Challenge &#8211; 2"},"content":{"rendered":"\n<p>How many pairs of positive integers $(x,y)$ are there such that $x&lt;y$ and $\\dfrac{x^2+y^2}{x+y}$ is an integer which is a divisor of $2835$? <em>BIMC 2018<\/em><\/p>\n\n\n\n<button onclick=\"toggle_visibility('nt-chall-2-solution')\">Click for the solution<\/button>\n<div style=\"display:none\" id=\"nt-chall-2-solution\">\n\n\n\n<p><strong>Solution<\/strong>: Based on the result of Number Theory Challenge &#8211; 1, if $x^2+y^2=pq$, where $p=4k+3$ is a prime number, we have $x\\equiv y\\equiv 0\\pmod{p}$. Therefore, if $(x,y)$ is a solution for $\\dfrac{x^2+y^2}{x+y}$ as an integer, then $(\\dfrac{x}{p},\\dfrac{y}{p})$ is also a solution.<\/p>\n\n\n\n<p>Because $2835=3^{4}\\cdot 5\\cdot 7$, and $3$ and $7$ are prime numbers in the format of $4k+3$, therefore, if we find all solutions for $$\\dfrac{x^2+y^2}{x+y}=q$$ where $q$ is not divisible by $3$ nor $7$, we can find all solutions. Therefore, we only need to consider two cases, where $q=1$ or $q=5$.<\/p>\n\n\n\n<p>Case 1: if $q=1$, we have $\\dfrac{x^2+y^2}{x+y}=1$, i.e. $$(2x-1)^2+(2y-1)^2=2$$ The above equation have no integer solution with $0&lt;x&lt;y$.<\/p>\n\n\n\n<p>Case 2: if $q=5$, we have $\\dfrac{x^2+y^2}{x+y}=5$, i.e. $$(2x-5)^2+(2y-5)^2=50$$ The two perfect square numbers on the left side of the above equation must be $1$ and $49$.<\/p>\n\n\n\n<p>Because $0&lt;x&lt;y$, we have $$2x-5=\\pm 1$$ $$2y-5=\\ \\ \\ 7$$<\/p>\n\n\n\n<p>which leads two solutions of $(x,y)$ as $(2, 6)$ or $(3,6)$.<\/p>\n\n\n\n<p>Therefore, the total number of solutions is $\\boxed{2\\cdot(4+1)\\cdot(1+1)=20}$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>How many pairs of positive integers $(x,y)$ are there such that $x&lt;y$ and $\\dfrac{x^2+y^2}{x+y}$ is an integer which is a divisor of $2835$? BIMC 2018 Click for the solution Solution: Based on the result of Number Theory Challenge &#8211; 1, &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2298\">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\/2298"}],"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=2298"}],"version-history":[{"count":12,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2298\/revisions"}],"predecessor-version":[{"id":2312,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2298\/revisions\/2312"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2298"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2298"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2298"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}