[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572390-105":3,"detail-sidebar-cat-1-en-105":85,"doc-detail-572390-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":84},105,"en","ch1-number-theory-cheat-sheet-key-algorithms","Ch.1 - Number Theory Cheat Sheet - Key Algorithms","","Number Theory Cheat Sheet summarizes core definitions and methods for working with integers and divisibility. It introduces integers, natural numbers, rationals, and reals, then formalizes divisibility and its properties. It explains the division algorithm, including quotient and remainder uniqueness, and shows how to use it to determine divisibility. It further derives gcd through the Euclidean algorithm and outlines the link to Bezout’s identity for expressing gcd as a linear combination.",{"@graph":14,"@context":77},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/general/","General",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/ch1-number-theory-cheat-sheet-key-algorithms/572390/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/ch1-number-theory-cheat-sheet-key-algorithms/572390.png","ImageObject",442,249,{"name":42,"@type":43},"Kurz","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69,73],{"name":60,"@type":61,"acceptedAnswer":62},"How is divisibility defined for integers, and what notation is used?","Question",{"text":63,"@type":64},"For integers a and b with a ≠ 0, b is divisible by a if there exists an integer k such that b = ka. This is written as a|b; if it does not hold, it is written as a ∤ b.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What does the division algorithm guarantee for two integers when the divisor is positive?",{"text":68,"@type":64},"If a and b are integers with b > 0, there exist unique integers q (quotient) and r (remainder) such that a = bq + r with 0 ≤ r \u003C b.",{"name":70,"@type":61,"acceptedAnswer":71},"How does the Euclidean algorithm compute the greatest common divisor (gcd)?",{"text":72,"@type":64},"Starting with two integers a ≥ b, the algorithm repeatedly applies the division algorithm to replace (a, b) with (b, r) until the remainder becomes 0. The last non-zero remainder is gcd(a, b).",{"name":74,"@type":61,"acceptedAnswer":75},"How does Bezout’s identity relate to gcd?",{"text":76,"@type":64},"Bezout’s identity states that for integers a and b (with b ≠ 0), there exist integers x and y such that gcd(a, b) = ax + by. The document also shows working backwards through the Euclidean algorithm to find x and y.","https://schema.org",{"og:url":32,"og:type":79,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":81,"canonical":32},"index,follow",{"doc_id":83,"site_id":7},572390,1792013013,{"code":4,"msg":86,"data":87},"success",[88,93,98,103,108,113,118,123,127],{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},11,"Presentations",90,"presentations",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},12,"Resumes",80,"resumes",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},14,"Invoices",70,"invoices",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},15,"Posters",60,"posters",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},16,"Social Media",50,"social-media",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},17,"Forms",40,"forms",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":120,"show_sort_weight":121,"slug":122},18,"Letters",30,"letters",{"id":124,"doc_module":22,"doc_module_name":25,"category_name":125,"show_sort_weight":55,"slug":126},21,"Paper Templates","papers-templates",{"id":128,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":4,"slug":129},158,"general-158",{"code":4,"msg":86,"data":131},{"doc_id":83,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":22,"category_id":128,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":22,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":4},2336478945635,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Number Theory Cheat Sheet  \nNumber theory is a branch of mathematics that is concerned with the study of the properties of numbers, and theinteresting and unexpected relationships between different sorts of numbers. Of course, these relationships alsoneed to be proved true. Number theory is used in cryptography.  \n# The division algorithm\n\nSystems of numbers can be referred to in different ways, it is important to know what each system refers to:  \n• The integers, ℤ = {… , −3, −2, −1, 0, 1, 2, 3, … }  \n• The natural numbers ℕ = {1, 2, 3, … }  \n• You may have already come across the real numbers, ℝ, or the rational numbers, ℚ .  \nIt is important to note that the set of natural numbers does not contain 0. The union of the set of natural numbers with 0is denoted ℕ0 .  \nDivisibility is a very important concept in number theory. Although you will have seen the concept of divisibility before andwill be very used to working with it, it is very useful to define it:  \n• If 􀜽 and 􀜾 are integers with 􀜽 ≠ 0, then 􀜾 is divisible by 􀜽 if there exists an integer 􀝇 such that 􀜾 = 􀝇􀜽 . If this  \nhappens, then we say that 􀜽 divides 􀜾 , which is denoted 􀜽|􀜾 . If 􀜽 does not divide 􀜾 then we write 􀜽 ∤ 􀜾This definition considers both positive and negative divisors. Divisibility also has the following properties:  \nFor any 􀜽, 􀜾 , 􀜿 ∈ ℤ, with 􀜽 ≠ 0:  \n• 􀜽|􀜽 (every integer divides itself)  \n• 􀜽|0 (0 is divisible by any integer)  \n• If 􀜽|􀜾 and 􀜾|􀜿 ⇒ 􀜽 |􀜿  \n• If 􀜽|􀜾 and 􀜽|􀜿 ⇒ 􀜽 |􀜾􀝊 + 􀜿􀝉 for all 􀝉, 􀝊 ∈ ℤ  \n• 􀜽|􀜾 ⇔ 􀜽􀝊 |􀜾􀝊 for all 􀝊 ∈ ℤ, 􀝊 ≠ 0  \n• If 􀜽 and 􀜾 are positive integers and 􀜽|􀜾 then 􀜽 ≤ 􀜾  \nExample 1: Prove that If 􀜽|􀜾 and 􀜾|􀜿 ⇒ 􀜽 |􀜿  \n\n| Use the fact that 􀜽 divides 􀜾 to rewrite 􀜾 in terms of 􀜽 and  \u003Cbr>􀜿 in terms of 􀜾   | 􀜾 = 􀝇􀜽, for some 􀝇 ∈ ℤ .  \u003Cbr>􀜿 = ℎ􀜾 , for some ℎ ∈ ℤ   |\n| --- | --- |\n| Substitute the equations into each other   | 􀜿 = ℎ(􀝇􀜽), or 􀜿 = (ℎ􀝇)􀜽  \u003Cbr>As ℎ, 􀝇 ∈ ℤ, ℎ × 􀝇 ∈ ℤ, as the set of integers are closedunder multiplication, so 􀜽|􀜿   |\n\nThe set of integers being closed under multiplication means that an integer multiplied by an integer will always give aninteger. The same is true for addition and subtraction with the set of integers but is not always true for division.Because of this, it’s helpful to have a better definition for division with the integers- the division algorithm allows us tofind a unique quotient and remainder for any two integers:  \nIf 􀜽 and 􀜾 are integers such that 􀜾 > 0, then there exists unique integers 􀝍 (which can be referred to as the quotient) and  \n􀝎 (remainder) such that 􀜽 = 􀜾􀝍 + 􀝎, with 0 ≤ 􀝎 \u003C 􀜾 . The process to find these integers is called the division algorithm:  \n1. Begin with your values of 􀜽 and 􀜾  \n2. Set 􀝍 equal to the greatest integer that is less than or equal to   \n3. Set 􀝎 = 􀜽 − 􀜾􀝍  \nNote that 􀜽 is divisible by 􀜾 if and only if 􀝎 = 0 .  \nExample 2: Find the quotient and remainder when 4649 is divided by 56  \n\n| Set the values of 􀜽 and 􀜾 .   | 􀜽 = 4649  \u003Cbr>􀜾 = 56   |\n| --- | --- |\n| Find the value of q   | 􀜽 = 83.0178􀜾  \u003Cbr>􀝍 = 83   |\n| Find the value of 􀝎   | 􀝎 = 􀜽 − 􀜾􀝍  \u003Cbr>􀝎 = 4649 − (56)(83)􀝎 = 1  \u003Cbr>4649 = 56(83) + 1   |\n\n# The Euclidean algorithm\n\nUsing the previous work on divisibility, we can write formal definitions of common divisors and greatest common divisors:  \n• For 􀜽, 􀜾 , 􀜿 ∈ ℤ, and 􀜿 ≠ 0, 􀜿 is a common divisor of 􀜽 and 􀜾 is 􀜿|􀜽 and 􀜿|􀜾  \n• The greatest common divisor, 􀝀, of 􀜽, 􀜾 ∈ ℤ, satisfies the following conditions:  \n- 􀝀|􀜽 and 􀝀|􀜾  \n- If 􀜿 is a common divisor of 􀜽 and 􀜾 , then 􀜿 ≤ 􀝀  \nThe greatest common divisor of 􀜽 and 􀜾 can also be denoted gcd (􀜽, 􀜾)  \nAs you will have seen before, the greatest common divisors can be found by writing the number as a product of primefactors, but this is an inefficient method for large numbers. The Euclidean algorithm provides an iterative method forfinding the greatest common divisor of two integers:  \n1. Given your two integers, denote them 􀜽 and 􀜾 such that 􀜽 ≥ 􀜾  \n2. Use the division algorithm to find integers 􀝍1 and 􀝎1 such t","cbCaic8C1MwOy81N","https://ap.wps.com/l/cbCaic8C1MwOy81N","pdf",277237,"English","# The division algorithm\n## Divisibility definitions and properties\n## Quotient and remainder\n# The Euclidean algorithm\n## Common divisors and greatest common divisor\n## Iterative computation of gcd\n## Bezout’s identity and linear combinations","[{\"question\":\"How is divisibility defined for integers, and what notation is used?\",\"answer\":\"For integers a and b with a ≠ 0, b is divisible by a if there exists an integer k such that b = ka. This is written as a|b; if it does not hold, it is written as a ∤ b.\"},{\"question\":\"What does the division algorithm guarantee for two integers when the divisor is positive?\",\"answer\":\"If a and b are integers with b \\u003e 0, there exist unique integers q (quotient) and r (remainder) such that a = bq + r with 0 ≤ r \\u003c b.\"},{\"question\":\"How does the Euclidean algorithm compute the greatest common divisor (gcd)?\",\"answer\":\"Starting with two integers a ≥ b, the algorithm repeatedly applies the division algorithm to replace (a, b) with (b, r) until the remainder becomes 0. The last non-zero remainder is gcd(a, b).\"},{\"question\":\"How does Bezout’s identity relate to gcd?\",\"answer\":\"Bezout’s identity states that for integers a and b (with b ≠ 0), there exist integers x and y such that gcd(a, b) = ax + by. The document also shows working backwards through the Euclidean algorithm to find x and y.\"}]","Ch.1 - Number Theory Cheat Sheet - Key Algorithms | PDF",1791927113]