{"id":481,"date":"2020-05-17T18:30:54","date_gmt":"2020-05-17T18:30:54","guid":{"rendered":"http:\/\/blog.newtonchineseschool.org\/lilijia\/?p=481"},"modified":"2020-05-17T22:00:50","modified_gmt":"2020-05-17T22:00:50","slug":"ch-13-divisibility-rules","status":"publish","type":"post","link":"https:\/\/blog.newtonchineseschool.org\/lilijia\/2020\/05\/17\/ch-13-divisibility-rules\/","title":{"rendered":"AoPS Number Theory Ch.13 Divisibility Rules"},"content":{"rendered":"\n<p>Congratulations to Jason, Julia, Alex, Susan, and Gioia on the Kahoot! Game today:)<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/WeChat-Image_20200517142235.png\" alt=\"\" class=\"wp-image-482\" width=\"342\" height=\"267\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/WeChat-Image_20200517142235.png 882w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/WeChat-Image_20200517142235-300x234.png 300w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/WeChat-Image_20200517142235-768x600.png 768w\" sizes=\"(max-width: 342px) 100vw, 342px\" \/><\/figure><\/div>\n\n\n\n<p><strong>Lesson review:<\/strong><\/p>\n\n\n\n<ul><li>Let m be a nonnegative integer. Every nonnegative integer is congruent modulo 2^m, modulo 5^m, and modulo 10^m to its last m digits.<\/li><li>Every nonnegative integer is congruent modulo 3, modulo 9 to the sum of its digits.<\/li><li>Every nonnegative integer is congruent modulo 11 to the alternative sum of its digits.<\/li><\/ul>\n\n\n\n<ul><li>2n\u21cbunits digit is even<\/li><li>5n\u21cbunits digit is 5 or 0<\/li><li>10n\u21cbunits digit is 0<\/li><\/ul>\n\n\n\n<ul><li>k*2^m \u21cb its last m digits themselves form an integer that is a multiple of 2^m<\/li><li>k*5^m \u21cb its last m digits themselves form an integer that is a multiple of 5^m<\/li><li>k*10^m \u21cb its last m digits are all 0.<\/li><\/ul>\n\n\n\n<ul><li>3n\u21cb the sum of its digit is 3k<\/li><li>9n\u21cb the sum of its digit is 9k<\/li><li>11n\u21cb the alternating sum of its digit is 11k<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"742\" height=\"611\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/xdivisibility-rules.png.pagespeed.ic_.CuwF9MaYq7.png\" alt=\"\" class=\"wp-image-485\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/xdivisibility-rules.png.pagespeed.ic_.CuwF9MaYq7.png 742w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/xdivisibility-rules.png.pagespeed.ic_.CuwF9MaYq7-300x247.png 300w\" sizes=\"(max-width: 742px) 100vw, 742px\" \/><\/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-0315-768x1024.jpg\" alt=\"\" class=\"wp-image-483\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0315-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0315-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0315.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-0316-768x1024.jpg\" alt=\"\" class=\"wp-image-484\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0316-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0316-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/05\/IMG-0316.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<p><strong>Extra Resource:<\/strong><\/p>\n\n\n\n<ul><li>Divisibility rules for 2-12<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Divisibility rules: 2 to 12\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Y1pAKJ4rf-M?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<ul><li>Divisibility rule for 11<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"SIMPLE Divisibility By 11 Test - The Alternating Sum Of Digits\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/digoW9eIMw4?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<ul><li>Divisibility rule for 9<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Divisibility rules for 9\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/PxgdsvdAqGk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p><strong>Homework:<\/strong><\/p>\n\n\n\n<ul><li>Pg. 252 Ex.13.2.1 (a,c), 13.2.2(c,f), 13.2.3(a,f), 13.2.4<\/li><li>Pg. 255 Ex.13.3.2, 13.3.4, 13.3.6<\/li><li>Pg. 257 Ex. 13.15 to 13.19, 13.25 <\/li><\/ul>\n\n\n\n<p>No School next week. Enjoy the Memorial Day Holiday, and see you all on May 31st!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Congratulations to Jason, Julia, Alex, Susan, and Gioia on the Kahoot! Game today:) Lesson review: Let m be a nonnegative integer. Every nonnegative integer is congruent modulo 2^m, modulo 5^m, and modulo 10^m to its last m digits. Every nonnegative integer is congruent modulo 3, modulo 9 to the sum of its digits. Every nonnegative [&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\/481"}],"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=481"}],"version-history":[{"count":2,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/481\/revisions"}],"predecessor-version":[{"id":492,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/481\/revisions\/492"}],"wp:attachment":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/media?parent=481"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/categories?post=481"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/tags?post=481"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}