{"id":503,"date":"2020-06-07T18:26:12","date_gmt":"2020-06-07T18:26:12","guid":{"rendered":"http:\/\/blog.newtonchineseschool.org\/lilijia\/?p=503"},"modified":"2020-06-07T22:46:25","modified_gmt":"2020-06-07T22:46:25","slug":"aops-number-theory-ch-14ii-linear-congruences","status":"publish","type":"post","link":"https:\/\/blog.newtonchineseschool.org\/lilijia\/2020\/06\/07\/aops-number-theory-ch-14ii-linear-congruences\/","title":{"rendered":"AoPS Number Theory: Ch. 14(II) Linear Congruences"},"content":{"rendered":"\n<p>Lesson review:<\/p>\n\n\n\n<ul><li>a linear congruence equation<\/li><li>an inverse: b*b^(-1) is congruent to 1 (mod m)<\/li><li>only if gcd (m,r)=1, otherwise r has no reverse modulo m.<\/li><li>if ak is congruent to bk (mod mk), then a is congruent to b (mod m)<\/li><li>if ac is congruent to bc (mod m), then a is congruent to b (mod m\/gcd(m,c))<\/li><li>many remainder problems are just linear congruences.<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" width=\"768\" height=\"1024\" src=\"http:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0439-768x1024.jpg\" alt=\"\" class=\"wp-image-504\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0439-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0439-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0439.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\/06\/IMG-0440-768x1024.jpg\" alt=\"\" class=\"wp-image-505\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0440-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0440-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0440.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\/06\/IMG-0441-768x1024.jpg\" alt=\"\" class=\"wp-image-506\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0441-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0441-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0441.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\/06\/IMG-0444-1-768x1024.jpg\" alt=\"\" class=\"wp-image-510\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0444-1-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0444-1-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0444-1.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\/06\/IMG-0445-768x1024.jpg\" alt=\"\" class=\"wp-image-511\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0445-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0445-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0445.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\/06\/IMG-0446-768x1024.jpg\" alt=\"\" class=\"wp-image-512\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0446-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0446-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0446.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\/06\/IMG-0448-768x1024.jpg\" alt=\"\" class=\"wp-image-513\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0448-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0448-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0448.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\/06\/IMG-0449-768x1024.jpg\" alt=\"\" class=\"wp-image-514\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0449-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0449-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0449.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\/06\/IMG-0450-768x1024.jpg\" alt=\"\" class=\"wp-image-515\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0450-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0450-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0450.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\/06\/IMG-0454-768x1024.jpg\" alt=\"\" class=\"wp-image-517\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0454-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0454-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0454.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\/06\/IMG-0455-768x1024.jpg\" alt=\"\" class=\"wp-image-518\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0455-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0455-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0455.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\/06\/IMG-0457-768x1024.jpg\" alt=\"\" class=\"wp-image-519\" srcset=\"https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0457-768x1024.jpg 768w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0457-225x300.jpg 225w, https:\/\/blog.newtonchineseschool.org\/lilijia\/files\/2020\/06\/IMG-0457.jpg 1536w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/figure>\n\n\n\n<p>Extra Resource<\/p>\n\n\n\n<ul><li><a href=\"https:\/\/mathworld.wolfram.com\/LinearCongruenceEquation.html\">https:\/\/mathworld.wolfram.com\/LinearCongruenceEquation.html<\/a><\/li><li><a href=\"https:\/\/www.youtube.com\/watch?v=4-HSjLXrfPs\">https:\/\/www.youtube.com\/watch?v=4-HSjLXrfPs<\/a><\/li><li><a href=\"https:\/\/math.libretexts.org\/Bookshelves\/Combinatorics_and_Discrete_Mathematics\/Book%3A_Elementary_Number_Theory_(Raji)\/03%3A_Congruences\/3.03%3A_Linear_Congruences\">https:\/\/math.libretexts.org\/Bookshelves\/Combinatorics_and_Discrete_Mathematics\/Book%3A_Elementary_Number_Theory_(Raji)\/03%3A_Congruences\/3.03%3A_Linear_Congruences<\/a><\/li><\/ul>\n\n\n\n<p>Homework<\/p>\n\n\n\n<ul><li>Pg. 270 Ex.14.3.1, 14.3.3,&nbsp;<\/li><li>Pg. 275 Ex.14.4.1, 14.4.4, 14.4.5<\/li><li>Pg. 278 Ex. 14.20, 14.27, 14.29&nbsp; <strong>to <\/strong>14.31 <\/li><\/ul>\n\n\n\n<p>Feel free to reach out if you have any questions in this Chapter. We will learn Ch.15 next Sunday, and have our final exam on 6\/21. <\/p>\n\n\n\n<p>Make sure you study hard and start your review as early as possible:)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Lesson review: a linear congruence equation an inverse: b*b^(-1) is congruent to 1 (mod m) only if gcd (m,r)=1, otherwise r has no reverse modulo m. if ak is congruent to bk (mod mk), then a is congruent to b (mod m) if ac is congruent to bc (mod m), then a is congruent to [&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\/503"}],"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=503"}],"version-history":[{"count":2,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/503\/revisions"}],"predecessor-version":[{"id":520,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/posts\/503\/revisions\/520"}],"wp:attachment":[{"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/media?parent=503"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/categories?post=503"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.newtonchineseschool.org\/lilijia\/wp-json\/wp\/v2\/tags?post=503"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}