{"id":35982,"date":"2026-01-10T18:40:43","date_gmt":"2026-01-10T18:40:43","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35982"},"modified":"2026-01-10T18:40:43","modified_gmt":"2026-01-10T18:40:43","slug":"the-euclidean-division-algorithm","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/the-euclidean-division-algorithm\/","title":{"rendered":"The Euclidean Division Algorithm"},"content":{"rendered":"<style>\np, ul, ol{\n  text-align: justify;\n}\nh1{\n  text-align:center;\n  text-transform: uppercase;\n}\nh2{\n  text-align:center;\n  text-transform: uppercase;\n  font-size:24pt;\n}\nh3{\n  text-align: center;\n  text-transform: uppercase;\n  font-size: 24px !important;\n}\n<\/style>\n<h1>The Division Algorithm<\/h1>\n<p style=\"text-align:center;\"><em>In this class we will develop the <strong>division algorithm<\/strong> as the principle that formalizes, for integers, the unique decomposition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>. The <strong>existence<\/strong> of the quotient and the remainder is first proved, and then their <strong>uniqueness<\/strong>. Finally, the meaning of the remainder is interpreted, the theory is linked to the <strong>long division algorithm<\/strong> as a computational procedure, and its natural connection with modular arithmetic and computational applications is anticipated.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives<\/strong><\/p>\n<p>Upon completing this class, the student will be able to:<\/p>\n<ol>\n<li><strong>Identify<\/strong> the basic concepts and roles of integer division (dividend, divisor, quotient, remainder) and the notion of divisibility as the exact case.<\/li>\n<li><strong>Explain<\/strong> the statement of the Euclidean division algorithm\/theorem, including the conditions that determine the remainder (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>) and their purpose in avoiding ambiguities.<\/li>\n<li><strong>Apply<\/strong> the Euclidean decomposition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> to determine <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> in concrete examples, verifying the bound on the remainder.<\/li>\n<li><strong>Analyze<\/strong> the treatment of cases according to the sign of the dividend and the divisor, justifying why the Euclidean convention maintains a nonnegative remainder.<\/li>\n<li><strong>Execute<\/strong> the long division algorithm as a mechanical procedure based on positional representation to obtain <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span>.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>TABLE OF CONTENTS<\/u>:<\/b><br \/>\n<a href=\"#1\">Division and divisibility<\/a><br \/>\n<a href=\"#2\">Euclidean division theorem<\/a><br \/>\n<a href=\"#3\">Proof of the Euclidean division theorem<\/a><br \/>\n<a href=\"#4\">Interpretation of the quotient and the remainder<\/a><br \/>\n<a href=\"#5\">Long division algorithm<\/a><br \/>\n<a href=\"#6\">Conclusion<\/a><br \/>\n<a href=\"#7\">Proposed and solved exercises<\/a>\n<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Division and divisibility<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/nlER6VzrGwA?si=79gpsAzwc7wcWEVn\" 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><\/center><\/p>\n<p>In arithmetic, divisibility describes the \u201cexact\u201d case: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> means that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> can be written exactly as a multiple of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. However, although divisibility does not always hold between two arbitrary integers, we can still ask \u201chow many times one number fits into another and, if it does not fit exactly, what is left over.\u201d This is the context in which the <strong>division algorithm<\/strong> arises, guaranteeing that any integer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> can be written as a multiple of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> plus a \u201cremainder.\u201d<\/p>\n<p>As an example, consider <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span>. It is clear that 3 does not divide 16. Nevertheless, it is possible to find a quotient and a remainder for the division operation, because<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">16 = 5\\cdot 3 + 1<\/span>\n<figure id=\"attachment_35775\" aria-describedby=\"caption-attachment-35775\" style=\"width: 1492px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg\" alt=\"A group of 16 boxes have been separated into groups of 3; as a result, 5 groups of boxes are obtained and 1 remains\" width=\"1492\" height=\"230\" class=\"size-full wp-image-35775 lazyload\" \/><figcaption id=\"caption-attachment-35775\" class=\"wp-caption-text\"><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg\" alt=\"A group of 16 boxes have been separated into groups of 3; as a result, 5 groups of boxes are obtained and 1 remains\" width=\"1492\" height=\"230\" class=\"size-full wp-image-35775 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg 1492w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-300x46.jpg 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-1024x158.jpg 1024w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-768x118.jpg 768w\" sizes=\"(max-width: 1492px) 100vw, 1492px\" \/><\/noscript> A group of 16 boxes have been separated into groups of 3; as a result, 5 groups of boxes are obtained and 1 remains<\/figcaption><\/figure>\n<p>Consequently, when dividing <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span> by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span> (where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> is the <strong>dividend<\/strong> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> is the <strong>divisor<\/strong>), represented by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\/3<\/span><\/span> or <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\div 3<\/span><\/span>, the <strong>quotient<\/strong> obtained is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=5<\/span><\/span> and the <strong>remainder<\/strong> is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>. In general, the <em>division algorithm<\/em> states that, for any integers <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> (divisor) and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> (dividend) with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, there exist unique integers <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> such that<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r,\\qquad 0\\le r&lt;|a|<\/span>\n<p>In this example, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16=3\\cdot 5+1<\/span><\/span>, and the condition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> is satisfied because <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 1&lt;3<\/span><\/span>. This formulation avoids ambiguities when the divisor (or the dividend) may be negative and guarantees that the remainder is always nonnegative and strictly less than the absolute value of the divisor.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Euclidean Division Theorem<\/h2>\n<p>The result of applying the division algorithm is what is known as <strong>Euclidean division<\/strong> and is based on the following result.<\/p>\n<p><b>Theorem:<\/b> 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>. Then there exist unique <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> such that<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r,\\qquad 0\\le r&lt;|a|.\n\n<\/span>\n<p>The integer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> is called the <strong>quotient<\/strong> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> is called the <strong>remainder<\/strong> of the division of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Proof of the Euclidean Division Theorem<\/h2>\n<p>This proof is divided into two parts: first, the existence of the quotient and the remainder is established; and then, given their existence, their uniqueness is shown.<\/p>\n<h3>Existence<\/h3>\n<p>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> and define <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span>, so that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d&gt;0<\/span><\/span>. We will first prove the case <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>: we will show that there exist <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>. We will then address the case <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span>. Finally, we will replace <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d<\/span><\/span> by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> to obtain the form <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>.<\/p>\n<p>For <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>, we establish the proposition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span>: \u201cthere exist <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>.\u201d We will prove by induction that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> holds for all <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>.<\/p>\n<p><strong>Base case:<\/strong> For <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>, taking <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=0<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span> yields <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=d\\cdot 0+0<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 0&lt;d<\/span><\/span>. Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(0)<\/span><\/span> is true.<\/p>\n<p><strong>Inductive step:<\/strong> Let <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\ge 0<\/span><\/span> and suppose that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k)<\/span><\/span> is true. Then there exist <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> such that<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk=dq+r,\\qquad 0\\le r&lt;d.\n\n<\/span>\n<p>Adding <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span> to both sides yields<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq+(r+1).\n\n<\/span>\n<p>Moreover, from <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span> it follows that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1\\le d<\/span><\/span>. Consequently, only the cases <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span> or <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span> can occur.<\/p>\n<p><b>Case 1:<\/b> If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span>, we define <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=r+1<\/span><\/span>. Then<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq&#039;+r&#039;,\\qquad 0\\le r&#039;&lt;d.\n\n<\/span>\n<p><b>Case 2:<\/b> If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq+d=d(q+1)+0<\/span><\/span>. We define <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q+1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=0<\/span><\/span>, and we have<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq&#039;+r&#039;,\\qquad 0\\le r&#039;&lt;d.\n\n<\/span>\n<p>In both cases we have constructed integers <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;,r&#039;<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq&#039;+r&#039;<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&#039;&lt;d<\/span><\/span>. Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k+1)<\/span><\/span> is true. We conclude by induction that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> holds for all <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>.<\/p>\n<p>Now consider the case <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span>. Then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b&gt;0<\/span><\/span>. Applying the previous result to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b<\/span><\/span>, there exist <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> such that<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n-b=dq+r,\\qquad 0\\le r&lt;d.\n\n<\/span>\n<p>Multiplying by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span> yields<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=-dq-r.\n\n<\/span>\n<p>If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span>, it suffices to take <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=0<\/span><\/span>, so that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>.<\/p>\n<p>If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&gt;0<\/span><\/span>, we define <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q-1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=d-r<\/span><\/span>. Since <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;r&lt;d<\/span><\/span>, we have <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;d-r&lt;d<\/span><\/span>, that is, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>. Moreover,<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\ndq_1+r_1=d(-q-1)+(d-r)=-dq-d+d-r=-dq-r=b.\n\n<\/span>\n<p>Consequently, for every <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span> there exist <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1,r_1\\in\\mathbb{Z}<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>.<\/p>\n<p>Finally, recall that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span>. If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=a<\/span><\/span> and the equality <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> can be written as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=aq_1+r_1<\/span><\/span>, with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;|a|<\/span><\/span>.<\/p>\n<p>If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&lt;0<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=-a<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1=(-a)q_1+r_1=a(-q_1)+r_1<\/span><\/span>. Defining <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-q_1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r_1<\/span><\/span>, we obtain<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r,\\qquad 0\\le r&lt;|a|.\n\n<\/span>\n<p>This completes the proof of the existence of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> for any <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>.<\/p>\n<h3>Uniqueness<\/h3>\n<p>Suppose that there exist <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,q&#039;,r,r&#039;\\in\\mathbb{Z}<\/span><\/span> such that<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r=q&#039;a+r&#039;,\\qquad 0\\le r,r&#039;&lt;|a|.\n\n<\/span>\n<p>Subtracting both expressions yields<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\na(q-q&#039;)=r&#039;-r.\n\n<\/span>\n<p>Since <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r,r&#039;&lt;|a|<\/span><\/span>, it follows that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|r&#039;-r|&lt;|a|<\/span><\/span>. On the other hand, the left-hand side is a multiple of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, and therefore also a multiple of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span>. The only integer multiple of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> with absolute value strictly less than <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span>; indeed, if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a| m \\neq 0<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m\\in\\mathbb{Z}<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\big| |a| m \\big| \\ge |a| <\/span><\/span>. Consequently, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;-r=0<\/span><\/span>, that is, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r&#039;<\/span><\/span>, and substituting into <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> yields <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=q&#039;<\/span><\/span>. Hence the decomposition is unique.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Interpretation of the Quotient and the Remainder<\/h2>\n<ul>\n<li style=\"text-align: justify;\">\n    <strong>What \u201cremainder\u201d means.<\/strong> If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=b-qa<\/span><\/span> is what \u201cremains\u201d after removing from <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> the multiple <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">qa<\/span><\/span>. In other words, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> is the residue obtained by fitting <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> to the grid of multiples of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>.\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>Why it is required that<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>. Without this condition, the pair <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(q,r)<\/span><\/span> would not be unique. Indeed, if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>, then also <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=a(q+1)+(r-a)<\/span><\/span>. The condition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> forces the remainder to lie in a fixed window <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,2,\\dots,|a|-1\\}<\/span><\/span>, thereby preventing the infinitely many \u201crelabelings\u201d of the same number.\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>What happens if<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>is negative.<\/strong> The theorem remains valid without changes: the remainder stays nonnegative and bounded by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span>. This is relevant because some programming languages use truncation toward zero and may produce negative remainders, whereas in mathematics the convention <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> is adopted so that the remainder is a \u201cstandard\u201d representative.<\/p>\n<p>    Moreover, when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span>, the quotient coincides with the floor:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nq=\\left\\lfloor \\frac{b}{a}\\right\\rfloor,\\qquad r=b-a\\left\\lfloor \\frac{b}{a}\\right\\rfloor.\n\n<\/span>\n<\/li>\n<\/ul>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Long Division Algorithm<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ABw4QYdD6Pg?si=eJiA7P6cD7RdlYYN\" 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><\/center><\/p>\n<p>Starting from Euclidean division, it is possible to implement the long division algorithm by taking advantage of the positional representation of numbers. This algorithm is a computational technique that allows one to quickly find the values <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> when one wishes to compute <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\div a<\/span><\/span> with <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\">a\\neq 0<\/span><\/span>.<\/p>\n<p>To describe it, we first review some examples that illustrate the process and the possible situations that may arise during the execution of the algorithm.<\/p>\n<ol>\n<li>\n<p>Suppose that we want to compute <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">57\\div 4<\/span><\/span>. Here 57 is the <strong>dividend<\/strong> and 4 is the <strong>divisor<\/strong>. To do so, we perform the following sequence of calculations:<\/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; \\color{red}57 \\div 4 =\\color{black}  \\\\ &amp; \\text{Write the division operation.} \\\\ \\\\\n\n(2) &amp; \\color{red}5&#039;\\color{black}7 \\div 4 =  \\\\ &amp; \\text{Separate the first prefix (from the left) of the dividend that}\\\\\n\n&amp;\\text{is greater than or equal to the divisor, in this case 5.} \\\\ \\\\\n\n(3) &amp; 5&#039;7 \\div 4 = \\color{red}1\\color{black} \\\\ &amp; \\text{Think of the greatest number that, when multiplied by $4$,}\\\\\n\n&amp;\\text{is less than or equal to $5$, and write it to the right of the equality. It is 1.} \\\\ \\\\\n\n(4) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;7 \\div 4 &amp;= 1 \\\\ \\color{red}-|\\underline{4}\\color{black} &amp; \\\\ \\color{red}\\phantom{-|}1\\color{black} &amp;  \\end{array} \\\\ &amp; \\text{Multiply the result by the divisor and subtract it from the selected value.} \\\\ \\\\\n\n(5) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;\\color{red}7&#039;\\color{black} \\div 4 &amp;= 1 \\\\ -|\\underline{4} &amp; \\\\ \\phantom{-|}1\\color{red}7\\color{black} &amp;  \\end{array} \\\\ &amp; \\text{Select and \u201cbring down\u201d the next digit.} \\\\ \\\\\n\n(6) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;7&#039; \\div 4 &amp;= 1\\color{red}4\\color{black} \\\\ -|\\underline{4} &amp; \\\\ \\phantom{-|}17 &amp; \\\\ \\color{red}-|\\underline{16}\\color{black} &amp; \\\\ \\color{red}\\phantom{-|}1\\color{black}  \\end{array} \\\\ &amp; \\text{Repeat the sequence with the last number obtained.}\n\n\\end{array}\n\n<\/span>\n<p>The algorithm ends when there are no more digits to \u201cbring down,\u201d yielding as a result the quotient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=14<\/span><\/span> and the remainder <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>. In particular, the remainder will always be smaller than the divisor.<\/p>\n<\/li>\n<li>\n<p>Suppose that we want to compute <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">132\\div 5<\/span><\/span>. To do so, we perform the following sequence of calculations:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(1) &amp;1&#039;32 \\div 5 = \\\\\n\n&amp; \\text{Separate the first prefix of the dividend and look for a natural number that,}\\\\\n\n&amp; \\text{when multiplied by 5, is less than or equal to it. If this is not possible,}\\\\\n\n&amp; \\text{incorporate the next digit until it becomes possible.} \\\\ \\\\\n\n(2) &amp;\\begin{array}{ll}\\phantom{-|}13&#039;2 \\div 5 &amp;= 2 \\\\ -|\\underline{10} &amp; \\\\ \\phantom{-|1}3 &amp; \\end{array}\\\\\n\n&amp; \\text{Once the previous step works, execute the algorithm normally.} \\\\ \\\\\n\n(3) &amp;\\begin{array}{ll}\\phantom{-|}13&#039;2&#039; \\div 5 &amp;= 26 \\\\ -|\\underline{10} &amp; \\\\ \\phantom{-|1}32 &amp;  \\\\ \\phantom{\\,} -|\\underline{30} &amp; \\\\ \\phantom{-|30} 2 &amp;  \\end{array}\n\n\\end{array}<\/span>\n<\/li>\n<li>\n<p>If the dividend has more digits, the procedure is carried out in the same way. For example, when computing <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">3521\\div 12<\/span><\/span>, one obtains:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ll}\n\n\\phantom{-|}35&#039;2&#039;1&#039; \\div 12 &amp; = 293 \\\\\n\n-|\\underline{24} &amp; \\\\\n\n\\phantom{-|}112 &amp; \\\\\n\n-|\\underline{108} &amp; \\\\\n\n\\phantom{-|10}41 &amp; \\\\\n\n\\phantom{0}-|\\underline{36} &amp; \\\\\n\n\\phantom{10-|}5 &amp; \\\\\n\n\\end{array}<\/span>\n<\/li>\n<li>\n<p>If one of the numbers is negative, there are two acceptable ways to present the result. However, the standard form in number theory is the one in which the remainder is positive and therefore consistent with Euclidean division. For example, if we compute <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span>, the following possible developments are obtained, both correct as identities:<\/p>\n<ul>\n<li>\n        <b>With a negative remainder:<\/b> This is obtained by removing the sign at the beginning of the operation and restoring it at the end.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ll}\n\n\\phantom{-}59&#039;8&#039; \\div 21 &amp;= 28 \\\\\n\n-|\\underline{42} &amp; \\\\\n\n\\phantom{-|&#039;}178 &amp; \\\\\n\n\\phantom{\\,}-|\\underline{168} &amp; \\\\\n\n\\phantom{-|00}10 &amp; \\\\\n\n\\end{array}<\/span>\n<p>From this, we have:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = 21 \\times 28 + 10<\/span>\n<p>Then, if we multiply the entire equality by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span>, we obtain:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10<\/span>\n<p>Thus, in this representation, the result of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> is quotient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span> and remainder <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=-10<\/span><\/span>.<\/p>\n<p>And if we multiply the entire equality once again by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span>, we obtain:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = (-21)\\times(-28) + 10<\/span>\n<p>Thus, the result of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598\\div (-21)<\/span><\/span> is quotient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span> and remainder <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=10<\/span><\/span>.<\/p>\n<\/li>\n<li>\n        <b>With a positive remainder and consistent with Euclidean division:<\/b> From the previous development we have:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10<\/span>\n<p>Then, by adding <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=21-21<\/span><\/span> to the right-hand side of this equality, we obtain:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10 \\color{red}+ 21 - 21\\color{black} = 21\\times(-29) + 11<\/span>\n<p>Therefore, being consistent with Euclidean division, the result of computing <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> yields a quotient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-29<\/span><\/span> and the remainder <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=11<\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<p>With respect to the convention in Euclidean division, only the result with a positive remainder corresponds to the Euclidean quotient and remainder. However, both expressions are correct identities and may be useful in different contexts. Although the long division algorithm may produce negative remainders, this is practical for obtaining results mechanically and for performing algebraic manipulations without additional steps. On the other hand, results with a positive remainder, consistent with Euclidean division, allow one to fix a canonical representative of each residue class (for example, in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,\\dots,n-1\\}<\/span><\/span>), which facilitates the precise and unambiguous labeling of integers. It should be noted that other conventions for representatives also exist, such as symmetric residues, which are equally valid once the rule is fixed.<\/p>\n<\/li>\n<\/ol>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Conclusion<\/h2>\n<p>The division algorithm guarantees that every integer division can be expressed uniquely as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>, thereby establishing a standard criterion for interpreting the quotient and the remainder even when negative numbers are involved. The long division algorithm is nothing more than a practical implementation of this decomposition, supported by positional representation, and it allows one to compute <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> mechanically. Finally, this Euclidean form not only avoids ambiguities, but also connects naturally with modular arithmetic: the remainder <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> acts as a canonical representative of the residue class of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> modulo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, forming the basis of many techniques in number theory and in computational applications that will be addressed in future installments.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Proposed and Solved Exercises<\/h2>\n<h3>Euclidean division<\/h3>\n<ol>\n<li><strong>(solved)<\/strong> Find <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> such that\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137 = 9q + r,\\qquad 0\\le r\\lt 9.<\/span>\n<p><strong>Solution:<\/strong> Find the largest number that, when multiplied by <span class=\"katex-eq\" data-katex-display=\"false\">9<\/span>, does not exceed <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137<\/span><\/span>. We have <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">15\\times 9 = 135<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\times 9 = 144<\/span><\/span>; therefore, the desired quotient is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=15<\/span><\/span>. Now compute the remainder <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=137-9q<\/span><\/span> and verify that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 9<\/span><\/span>. Carrying out the verification, we obtain:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nr=137-9\\cdot 15=137-135=2.\n\n<\/span>\n<p>Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137=9\\cdot 15+2<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 2\\lt 9<\/span><\/span>.<\/p>\n<\/li>\n<li><strong>(solved)<\/strong> Find <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> such that\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025 = 37q + r,\\qquad 0\\le r\\lt 37.<\/span>\n<p><strong>Solution:<\/strong> Find the largest number that, when multiplied by <span class=\"katex-eq\" data-katex-display=\"false\">37<\/span>, does not exceed <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025<\/span><\/span>. Observe that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 54 = 1998<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 55 = 2035<\/span><\/span>; therefore, the desired quotient is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=54<\/span><\/span>. Now compute the remainder <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=2025-37q<\/span><\/span> and verify that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 37<\/span><\/span>. Carrying out the verification, we obtain:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nr=2025-37\\cdot 54=2025-1998=27.\n\n<\/span>\n<p>Therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025=37\\cdot 54+27<\/span><\/span> with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 27\\lt 37<\/span><\/span>.<\/p>\n<\/li>\n<li>Find <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> such that\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">745 = 23q + r,\\qquad 0\\le r\\lt 23.<\/span>\n<\/li>\n<li>Find <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> such that\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-314 = 11q + r,\\qquad 0\\le r\\lt 11.<\/span>\n<\/li>\n<li>Find <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> such that\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = (-21)q + r,\\qquad 0\\le r\\lt |-21|.<\/span>\n<\/li>\n<li><strong>Fixed remainder and search for integers.<\/strong> Let <span class=\"katex-eq\" data-katex-display=\"false\">a=12<\/span>.\n<p>(a) Describe the set of all integers <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> whose Euclidean division by <span class=\"katex-eq\" data-katex-display=\"false\">12<\/span> has remainder <span class=\"katex-eq\" data-katex-display=\"false\">r=5<\/span>.<\/p>\n<p>(b) Find the smallest integer <span class=\"katex-eq\" data-katex-display=\"false\">b\\gt 1000<\/span> that satisfies the above and determine the corresponding quotient <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span>.<\/p>\n<\/li>\n<\/ol>\n<h3>Long Division Algorithm<\/h3>\n<p>In each case, apply the long division algorithm to compute the quotient <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> and the remainder <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>. Convert the result to Euclidean division when the algorithm yields a negative remainder.\n<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">84\\div 6.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">197\\div 8.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1256\\div 7.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-3521\\div 12.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-98765\\div 24.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">845\\div -13.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-12345\\div -37.<\/span><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>The Division Algorithm In this class we will develop the division algorithm as the principle that formalizes, for integers, the unique decomposition with . The existence of the quotient and the remainder is first proved, and then their uniqueness. Finally, the meaning of the remainder is interpreted, the theory is linked to the long division [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35980,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":10,"footnotes":""},"categories":[567,1402],"tags":[],"class_list":["post-35982","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>The Euclidean Division Algorithm - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn the division algorithm and Euclidean division: existence and uniqueness of q and r, cases involving negative numbers, long division algorithm, examples, and exercises.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/en\/the-euclidean-division-algorithm\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Euclidean Division Algorithm\" \/>\n<meta property=\"og:description\" content=\"Learn the division algorithm and Euclidean division: existence and uniqueness of q and r, cases involving negative numbers, long division algorithm, examples, and exercises.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/the-euclidean-division-algorithm\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2026-01-10T18:40:43+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2026\/01\/divisioneuclidiana-1024x683.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"The Euclidean Division Algorithm\" \/>\n<meta name=\"twitter:description\" content=\"Learn the division algorithm and Euclidean division: existence and uniqueness of q and r, cases involving negative numbers, long division algorithm, examples, and exercises.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2026\/01\/divisioneuclidiana.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-division-algorithm\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-division-algorithm\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Euclidean Division Algorithm\",\"datePublished\":\"2026-01-10T18:40:43+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-division-algorithm\\\/\"},\"wordCount\":3086,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-division-algorithm\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/divisioneuclidiana.jpg\",\"articleSection\":[\"Mathematics\",\"Number Theory\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-division-algorithm\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-division-algorithm\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-euclidean-division-algorithm\\\/\",\"name\":\"The Euclidean Division Algorithm - 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