{"id":458,"date":"2020-05-11T01:13:07","date_gmt":"2020-05-11T01:13:07","guid":{"rendered":"http:\/\/blog.newtonchineseschool.org\/lilijia\/?p=458"},"modified":"2020-05-11T01:27:54","modified_gmt":"2020-05-11T01:27:54","slug":"aops-number-theory-ch-12-intro-to-modular-arithmetic","status":"publish","type":"post","link":"https:\/\/blog.newtonchineseschool.org\/lilijia\/2020\/05\/11\/aops-number-theory-ch-12-intro-to-modular-arithmetic\/","title":{"rendered":"AoPS Number Theory Ch.12 Intro to Modular Arithmetic"},"content":{"rendered":"\n<p><strong>Lesson review<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0193-768x1024.jpg\" alt=\"\" class=\"wp-image-459\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0193-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0193-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0193.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0195-768x1024.jpg\" alt=\"\" class=\"wp-image-460\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0195-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0195-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0195.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0196-768x1024.jpg\" alt=\"\" class=\"wp-image-461\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0196-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0196-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0196.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"666\" height=\"750\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/WeChat-Image_20200510193653.png\" alt=\"\" class=\"wp-image-462\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/WeChat-Image_20200510193653.png 666w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/WeChat-Image_20200510193653-266x300.png 266w\" sizes=\"(max-width: 666px) 100vw, 666px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0199-768x1024.jpg\" alt=\"\" class=\"wp-image-463\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0199-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0199-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0199.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0200-768x1024.jpg\" alt=\"\" class=\"wp-image-464\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0200-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0200-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0200.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0202-768x1024.jpg\" alt=\"\" class=\"wp-image-465\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0202-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0202-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0202.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0205-768x1024.jpg\" alt=\"\" class=\"wp-image-472\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0205-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0205-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0205.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0206-768x1024.jpg\" alt=\"\" class=\"wp-image-466\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0206-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0206-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0206.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0207-768x1024.jpg\" alt=\"\" class=\"wp-image-467\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0207-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0207-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0207.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0210-768x1024.jpg\" alt=\"\" class=\"wp-image-468\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0210-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0210-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0210.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0211-768x1024.jpg\" alt=\"\" class=\"wp-image-470\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0211-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0211-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0211.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0213-768x1024.jpg\" alt=\"\" class=\"wp-image-471\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0213-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0213-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0213.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<p><strong>Summary<\/strong><\/p>\n\n\n\n<ul><li>Modular system, modulus m, residue r<\/li><li>r is the modulo-m residue of n when n\\equiv r (mod m) and&nbsp; 0&lt;= r &lt; m<\/li><li>congruent or equivalent modulo m<\/li><li>The integers congruent to r modulo m make up a residue class (congruence classes or equivalence classes).<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image\"><img src=\"https:\/\/lh5.googleusercontent.com\/E4PeLBnPg0YPZztXKZYmdCIyGplOpn_6DkBwuVLCuXOejd-Ue7mPF86oFxQOvixG0oH0HXL_hnTpUr91T_OJXgT9sPPgxDnZfsVAYpiGVLmw197ApJ54YNDiraYF9Frh2uHzMvtk_S8\" alt=\"\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img src=\"https:\/\/lh4.googleusercontent.com\/iVGgMVvU6BC35hcugoisC4bCKEdWwdqRN3TjIhmS_FKCp8-OoCm-2ilvEm0xVA8sztl95Nl_wp5Sf3gPq28xlcgO8M7Oh18eXGHmxxS4_DKB9ZnssR4EfHI-2boFzG9bny0pASWdMZ4\" alt=\"\" \/><\/figure>\n\n\n\n<p><strong>Extra Resource<\/strong><\/p>\n\n\n\n<ul><li><a href=\"https:\/\/artofproblemsolving.com\/wiki\/index.php\/Modular_arithmetic\/Introduction\">https:\/\/artofproblemsolving.com\/wiki\/index.php\/Modular_arithmetic\/Introduction<\/a><\/li><li><a href=\"https:\/\/study.com\/academy\/lesson\/modular-arithmetic-rules-properties.html\">https:\/\/study.com\/academy\/lesson\/modular-arithmetic-rules-properties.html<\/a><\/li><li><a href=\"https:\/\/betterexplained.com\/articles\/fun-with-modular-arithmetic\/\">https:\/\/betterexplained.com\/articles\/fun-with-modular-arithmetic\/<\/a><\/li><li><a href=\"https:\/\/www.khanacademy.org\/math\/ap-calculus-bc\/bc-series-new\/bc-series-optional\/v\/deriving-geometric-series-sum-formula\">https:\/\/www.khanacademy.org\/math\/ap-calculus-bc\/bc-series-new\/bc-series-optional\/v\/deriving-geometric-series-sum-formula<\/a><\/li><\/ul>\n\n\n\n<p><strong>Homework<\/strong><\/p>\n\n\n\n<ul><li>p.221 Ex.12.2.1 (i,j,k,l), 12.2.3, 12.2.6<\/li><li>p.224 Ex.12.3.1, 12.3.2 <strong>to<\/strong> 12.3.4(a,c,e,f)<\/li><li>p.230 Ex.12.4.1 (a,c,e,f),12.4.2, 12.4.4<\/li><li>p.236 Ex.12.5.1(a,c,e,f), 12.5.2, 12.5.5<\/li><li>p.239 Ex.12.6.2(a), 12.6.5<\/li><li>p.242 Ex.12.26<\/li><\/ul>\n\n\n\n<p>Sorry that we have so much information for this chapter \ud83d\ude41 Please feel free to reach out if you have any questions. See you next Sunday!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Lesson review Summary Modular system, modulus m, residue r r is the modulo-m residue of n when n\\equiv r (mod m) and&nbsp; 0&lt;= r &lt; m congruent or equivalent modulo m The integers congruent to r modulo m make up a residue class (congruence classes or equivalence classes). Extra Resource https:\/\/artofproblemsolving.com\/wiki\/index.php\/Modular_arithmetic\/Introduction https:\/\/study.com\/academy\/lesson\/modular-arithmetic-rules-properties.html https:\/\/betterexplained.com\/articles\/fun-with-modular-arithmetic\/ https:\/\/www.khanacademy.org\/math\/ap-calculus-bc\/bc-series-new\/bc-series-optional\/v\/deriving-geometric-series-sum-formula Homework [&hellip;]<\/p>\n","protected":false},"author":157,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0},"categories":[4],"tags":[],"_links":{"self":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/458"}],"collection":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/users\/157"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/comments?post=458"}],"version-history":[{"count":2,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/458\/revisions"}],"predecessor-version":[{"id":474,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/458\/revisions\/474"}],"wp:attachment":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/media?parent=458"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/categories?post=458"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/tags?post=458"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}