{"id":35657,"date":"2025-12-26T18:56:32","date_gmt":"2025-12-26T18:56:32","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35657"},"modified":"2025-12-26T18:56:32","modified_gmt":"2025-12-26T18:56:32","slug":"divisibility","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/divisibility\/","title":{"rendered":"What is Divisibility?"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Divisibility<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n<strong>Divisibility<\/strong> is the true starting point of number theory because it turns the integers into a structured system: numbers are no longer viewed merely as \u201cquantities\u201d, but as elements that either fit together or do not. With a single relation, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, it is possible to express everything from simplification and factorization criteria to the core of certain procedures such as the Euclidean algorithm, which makes it possible to compute greatest common divisors in seconds even for large numbers. Moreover, it constitutes the technical foundation of ideas that appear repeatedly in applied mathematics and computation: congruences, modular arithmetic, validations, codes, and, later on, cryptography. Mastering divisibility is, in essence, learning to detect invisible patterns within the integers and to transform them into procedures that always work.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>Learning Objectives<\/b><br \/>\nAt the end of this set of notes, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the divisibility relation between integers.<\/li>\n<li><strong>Understand<\/strong> the definition of divisibility and its properties.<\/li>\n<li><strong>Develop<\/strong> mathematical proofs of results and theorems related to divisibility.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>TABLE OF CONTENTS<\/u><\/b><br \/>\n<a href=\"#1\">Definition of divisibility<\/a><br \/>\n<a href=\"#2\">Fundamental properties of divisibility<\/a><br \/>\n<a href=\"#3\">Proposed Exercises<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HLrwdLse18U?si=tDiiV02P7ppdb4xF\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/br><\/p>\n<h2>Definition of divisibility<\/h2>\n<p style=\"text-align: justify;\">\nThe informal idea of \u201c<em>a divides b<\/em>\u201d becomes precise when it is expressed as a relation between integers. We say that an integer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>divides<\/strong> an integer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> can be written as an exact multiple of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. This definition is the basis of the rest of these notes, because it transforms phrases such as \u201cfits exactly\u201d into a verifiable criterion.\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>Definition.<\/strong> Let <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. We say that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> divides <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>, and we write <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, if and only if there exists an integer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>. Otherwise, we write <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\nmid b<\/span><\/span>.\n<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b := (\\exists k \\in \\mathbb{Z})(b = ka )<\/span>\n<p style=\"text-align: justify;\">\nIn this definition, the number <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> is called the <strong>quotient<\/strong> (or factor) associated with the divisibility. For example, stating <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> is equivalent to stating that there exists <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span>; in this case, it suffices to take <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span>.\n<\/p>\n<h3>It is important to consider<\/h3>\n<ul>\n<li>\n    The condition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> is essential. If we attempted to allow <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span>, the divisibility condition would require the existence of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>. However, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span> for all <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>, so the only possibility would be <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>. In that case, there would be no <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u201cdetermined\u201d by the relation, since any <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> satisfies <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span>. In other words, the informal expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> becomes <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span>, which is undefined. To avoid this degeneration, in which the notion of quotient ceases to be meaningful, the condition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> is imposed. For this reason, the relation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> is not considered valid.\n<\/li>\n<li>\n        By contrast, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> is true for all <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, because it suffices to take <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, and the equality <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=a\\cdot 0<\/span><\/span> holds.\n    <\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\nFrom this definition, an equivalence follows that we will use repeatedly: stating that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> is the same as stating that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> belongs to the set of integer multiples of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, that is, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>, where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>. This way of writing emphasizes that divisibility is not a \u201ctrick\u201d, but rather a way of describing highly structured subsets within <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}<\/span><\/span>.<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h2>Fundamental properties of divisibility<\/h2>\n<ul>\n<li><strong>Reflexivity:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>.<br \/>\n<u>Proof<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1)&amp;\\vdash a=ka \\leftrightarrow k=1 &amp;\\text{; Multiplicative identity in $\\mathbb{Z}$}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; Existential introduction (1)}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; Definition of divisibility (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>Transitivity:<\/strong> if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\mid c<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>.\n<p><u>Proof<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1\\in\\mathbb{Z})(b=k_1a)  &amp;\\text{; Definition of divisibility, Assumption}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; Definition of divisibility, Assumption}\\\\\n\n(3)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})(b=k_1a \\wedge c=k_2b)  &amp;\\text{; $\\exists$-Compaction(1,2)}\\\\\n\n(4)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})(k_2b=k_1k_2a \\wedge c=k_2b)  &amp;\\text{; From(3)}\\\\\n\n(5)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})( c=k_1k_2a)  &amp;\\text{; From(4)}\\\\\n\n&amp;\\text{Algebra within the quantifier}&amp; \\\\\n\n(6)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k\\in\\mathbb{Z})( c=ka)  &amp;\\text{; From(5)}\\\\\n\n&amp;\\text{Closure of $\\mathbb{Z}$ under multiplication}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; Definition of divisibility (6)}\\\\\n\n(8)&amp; \\vdash (a\\mid b \\wedge  b\\mid c) \\rightarrow  a\\mid c  &amp;\\text{; $\\wedge$-ED(7)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>Compatibility with addition and subtraction:<\/strong> if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b+c)<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b-c)<\/span><\/span>.<br \/>\n<u>Proof<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1 \\in \\mathbb{Z})(b=k_1 a) &amp;\\text{; Definition of divisibility, Assumption}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; Definition of divisibility, Assumption}\\\\\n\n(3)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b=k_1 a \\wedge c=k_2 a) &amp;\\text{; $\\exists$-Compaction(1,2)}\\\\\n\n(4)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b+c= (k_1+k_2)a) &amp;\\text{; From(3)}\\\\\n\n&amp;\\text{Algebra within the quantifier.}&amp; \\\\\n\n(5)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k \\in \\mathbb{Z})(b+c= ka) &amp;\\text{; From(4)}\\\\\n\n&amp;\\text{Closure of $\\mathbb{Z}$ under addition.}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; Definition of divisibility (5)}\\\\\n\n(7)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b+c) &amp;\\text{; $\\wedge$-ED(6)}\\\\\n\n(8)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b-c= (k_1-k_2)a) &amp;\\text{; From(3)}\\\\\n\n&amp;\\text{Algebra within the quantifier.}&amp; \\\\\n\n(9)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists \\overline{k} \\in \\mathbb{Z})(b-c= \\overline{k}a) &amp;\\text{; From(8)}\\\\\n\n&amp;\\text{Closure of $\\mathbb{Z}$ under subtraction.}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; Definition of divisibility (9)}\\\\\n\n(11)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b-c) &amp;\\text{; $\\wedge$-ED(10)}\\\\\n\n(12)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow \\left(a\\mid (b+c) \\wedge a\\mid (b-c)\\right) &amp;\\text{;$\\wedge$-Introduction in the consequent(7,11) }\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>Compatibility with products:<\/strong> if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span> for all <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span>.<br \/>\n<u>Proof<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1)&amp; \\{a\\mid b\\}\\vdash (\\exists k\\in\\mathbb{Z})(b=ka) &amp;\\text{; Definition of divisibility, Assumption}\\\\\n\n(2)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists k\\in\\mathbb{Z})(cb=cka) &amp;\\text{; From(1), $\\forall$-Introduction (arbitrary c)}\\\\\n\n&amp;\\text{Algebra in }\\mathbb{Z}\\text{ within the existential quantifier.}&amp;\\\\\n\n(3)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists \\overline{k}\\in\\mathbb{Z})(cb=\\overline{k}a) &amp;\\text{; From(2), closure: }\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; Definition of divisibility (3)}\\\\\n\n(5)&amp; \\vdash a\\mid b \\rightarrow \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; ED(4)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<\/ul>\n<h3><b>Theorem:<\/b> bound on the divisor<\/h3>\n<p style=\"text-align: justify;\">\nIf <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\neq 0<\/span><\/span> and <span dir=\"ltr\"> <span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>.\n<\/p>\n<p><b>Proof:<\/b><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash b \\neq 0\n\n&amp; \\text{; Assumption} \\\\\n\n(2) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (b=ka)\n\n&amp; \\text{; Definition of divisibility, Assumption} \\\\\n\n(3) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (|b|=|k||a|)\n\n&amp; \\text{; Absolute value property, From(2)} \\\\\n\n(4) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (k\\neq 0 \\wedge |b|=|k||a|)\n\n&amp; \\text{; From(1,3)} \\\\\n\n(5) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (1\\le |k| \\wedge |b|=|k||a|)\n\n&amp; \\text{; From(4), if }k\\neq 0\\Rightarrow |k|\\ge 1 \\\\\n\n(6) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash |a|\\le |b|\n\n&amp; \\text{; From(5)} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>Proposed Exercises<\/h2>\n<ol>\n<li>Show that the theorem \u201cbound on the divisor\u201d is not necessarily true if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span><\/li>\n<li>Consider a set <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and a relation <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> on that set. If the elements <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> are such that <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> is related to <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> by means of <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span>, then it is written <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span><\/span>. The relation <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> is said to be a <strong>partial order<\/strong> on <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> if:\n<p>a)<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in A) (x\\rho x)<\/span><\/span>,<br \/>\nb) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y\\in A) ( (x\\rho y \\wedge y\\rho x) \\rightarrow x=y)<\/span><\/span><br \/>\nc) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y,z\\in A) ( (x\\rho y \\wedge y\\rho z) \\rightarrow x\\rho z)<\/span><\/span>.<\/p>\n<p>Prove that the divisibility relation is a partial order relation on the integers.<\/li>\n<li>Prove by induction that if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_1, a\\mid b_2, \\cdots, a\\mid b_n<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span> for any set <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span>. Also prove that if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span>, with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> can be written as a linear combination of those <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>. <\/li>\n<li>If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, show that the set <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> is a finite set.<\/li>\n<li>Consider a fixed <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span>, and let\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S=\\{d\\,:\\,d\\in\\mathbb{Z}^+ \\wedge d\\mid n\\}<\/span>\n<p>Prove:<\/p>\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d\\in S \\leftrightarrow n\/d\\in S<\/span><\/span><\/li>\n<li type=\"a\">If the elements of <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> are arranged in increasing order: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1=d_1 \\lt d_2 \\lt \\cdots \\lt d_t =n<\/span><\/span>, then the corresponding elements <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span> are in decreasing order.<\/li>\n<\/ol>\n<\/li>\n<li>Suppose that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}^+<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=c<\/span><\/span>. Prove that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>.<\/li>\n<li>An integer <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> is said to be even if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>, and odd if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>. Prove that the sum and the difference of:\n<ol>\n<li type=\"a\">two even numbers is an even number.<\/li>\n<li type=\"a\">two odd numbers is an even number.<\/li>\n<li type=\"a\">an even number and an odd number is an odd number.<\/li>\n<\/ol>\n<\/li>\n<li>If <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> is an odd integer different from <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span>, prove that <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> cannot divide two consecutive even numbers.<\/li>\n<li>Let <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> be such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>. Prove that <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> cannot divide either <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> or <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>.<\/li>\n<li>Suppose that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>. Prove that:\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall n \\in \\mathbb{Z})(a\\mid n) \\leftrightarrow a=\\pm 1<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall n \\in \\mathbb{Z})(n\\mid a) \\leftrightarrow a=0<\/span><\/span><\/li>\n<\/ol>\n<\/li>\n<li>Let <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> and suppose that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>. Show that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span> implies that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Divisibility Divisibility is the true starting point of number theory because it turns the integers into a structured system: numbers are no longer viewed merely as \u201cquantities\u201d, but as elements that either fit together or do not. With a single relation, , it is possible to express everything from simplification and factorization criteria to the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35642,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[567,1402],"tags":[],"class_list":["post-35657","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-number-theory"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>What is Divisibility? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u00bfQu\u00e9 es la divisibilidad en la teor\u00eda de n\u00fameros? 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