{"id":3412,"date":"2012-09-14T17:40:13","date_gmt":"2012-09-14T17:40:13","guid":{"rendered":"http:\/\/SchoolTutoring.com\/help\/?p=3412"},"modified":"2014-12-02T08:32:02","modified_gmt":"2014-12-02T08:32:02","slug":"divisibility-and-division-algorithm","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/divisibility-and-division-algorithm\/","title":{"rendered":"Divisibility and Division Algorithm"},"content":{"rendered":"

All positive and negative integers together zero form the set of integers which is usually denoted by Z<\/em><\/strong> or I<\/em><\/strong>. So, Z =<\/strong> {\u2026,-3,-2,-1,0,1,2,3,\u2026}.<\/strong> Let a <\/em><\/strong>and b<\/em><\/strong> be any two integers where a\u22600.<\/strong> If there exists an integer k <\/em><\/strong>such that a = bk<\/strong> then we say that b divides a<\/em><\/strong> and we write it as b|a<\/strong>. Here the symbol |<\/strong> denotes divides<\/em><\/strong>.<\/p>\n

So, we have b|a<\/strong> if and only if a=bk<\/strong> for some integer k<\/em><\/strong>.<\/p>\n

b|a\u00a0<\/strong> means b<\/strong> divides a<\/strong><\/em> or b<\/strong> is a factor of a<\/strong><\/em> or a<\/strong> is a multiple of b<\/strong><\/em>.<\/p>\n

Similarly we can write a|b<\/strong> if and only if b=ak<\/strong> for some integer k<\/em><\/strong>.<\/p>\n

a|b\u00a0<\/strong> means a<\/strong> divides b<\/strong><\/em> or a is a factor of b<\/strong><\/em> or b<\/strong> is a multiple of a<\/strong><\/em>.<\/p>\n

Example:<\/strong><\/p>\n

a)\u00a0\u00a0\u00a0\u00a0\u00a0 2|6<\/strong>, because 6= 2×3<\/strong>; Here we say 2<\/strong> is a factor of 6<\/strong> or 6<\/strong> is a multiple of 2<\/strong>.<\/p>\n

b)\u00a0\u00a0\u00a0\u00a0\u00a0 3<\/strong> does not divide 5<\/strong>.<\/p>\n

Properties of divisibility:<\/h5>\n

1)\u00a0\u00a0\u00a0\u00a0\u00a0 For any two integers a<\/strong> and b, a|b<\/strong> and b|a => a=\u00b1b.<\/strong><\/p>\n

2)\u00a0\u00a0\u00a0\u00a0\u00a0 Transitive property: If a|b<\/strong> and b|c<\/strong> then a|c.<\/strong><\/p>\n

3)\u00a0\u00a0\u00a0\u00a0\u00a0 If a|b<\/strong> and a|c<\/strong> then i) a|(b+c) ii) a|(b-c) iii) a|bc.<\/strong><\/p>\n

4)\u00a0\u00a0\u00a0\u00a0\u00a0 If a|b<\/strong> and x<\/strong> is any integer then a|bx<\/strong><\/p>\n

5)\u00a0\u00a0\u00a0\u00a0\u00a0 If a|b<\/strong> and c|d<\/strong> then ac|bd<\/strong><\/p>\n

6)\u00a0\u00a0\u00a0\u00a0\u00a0 If ac|bc<\/strong> then a|b.<\/strong><\/p>\n

Division algorithm:<\/h5>\n

In a normal division, it is known that the dividend can be written as the sum of the remainder and the product of divisor and quotient. The same thing is defined as division algorithm<\/em><\/strong> and is mathematically defined as follows.<\/p>\n

\u201cFor any two integers a<\/strong> and b<\/strong>, there exists integers q<\/strong> and r<\/strong> such that a=bq+r,<\/strong> where 0\u2264r<b.\u201d<\/strong><\/em><\/p>\n

Example:<\/strong><\/p>\n

We know that 36 \u00f7 5 <\/strong>gives 7<\/strong> as the quotient and 1<\/strong> as the remainder. So, by division algorithm we can write<\/p>\n

36 = 5×7 + 1<\/strong>.<\/p>\n

Still need help with Mathematics? Please read more about our Mathematics<\/a> tutoring services.<\/p>\n

SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Paso-Robles visit: Tutoring in Paso-Robles.<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

All positive and negative integers together zero form the set of integers which is usually denoted by Z or I. So, Z = {\u2026,-3,-2,-1,0,1,2,3,\u2026}. Let a and b be any two integers where a\u22600. If there exists an integer k such that a = bk then we say that b divides a and we write […]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[81,524,527,529,642,1154,1412],"class_list":["post-3412","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-algorithm","tag-divides","tag-divisible","tag-division","tag-factor","tag-multiple","tag-properties"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3412","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=3412"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3412\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=3412"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=3412"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=3412"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}