[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572391-105":53,"doc-detail-572391-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","number-theory-cheat-sheet","Number Theory Cheat Sheet","","Number theory cheat sheet covering core techniques for working with integers and their prime structure. Includes parity rules, perfect square properties and prime lists, methods for prime factorization and a division-based approach, and formulas for counting total factors of a natural number. Provides procedures to compute the greatest common factor (GCF) and least common multiple (LCM) via prime exponents, divisibility rules by multiple divisors, and guidance for common remainder problems using common multiples and complements, with Pythagorean triples examples.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/number-theory-cheat-sheet/572391/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/number-theory-cheat-sheet/572391.png","ImageObject",442,249,{"name":88,"@type":89},"Franzy","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How do parity rules determine even/odd results for basic operations?","Question",{"text":109,"@type":110},"The sheet states combinations such as even+even=even, even+odd=odd, and odd×odd=odd (with the other products listed similarly). Use these combinations to predict parity without full calculation.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What is the formula for the total number of positive factors from a prime factorization?",{"text":114,"@type":110},"For n with prime factorization n=p1^e1·p2^e2·…·pc^ec, the number of factors is (e1+1)(e2+1)…(ec+1). It also notes that a perfect square has an odd number of factors.",{"name":116,"@type":107,"acceptedAnswer":117},"How can you find the GCF and LCM using prime factorization?",{"text":118,"@type":110},"For the GCF, take the minimum exponent of each prime across the numbers; for the LCM, take the maximum exponent of each prime. The sheet provides an example using 24, 56, and 96 to compute GCF=8 and LCM=672.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572391,1792005721,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":76},2336478945519,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","# Number Theory Cheat Sheet\n\nParity Rules  \neven+even =evenodd+odd =eveneven +odd=oddeven×even =evenodd×odd =oddeven×odd =even  \n## Perfect Squares &Properties\n\nCommon primes:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97Common perfect squares:  \n1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400Factorizing a perfect square into prime powers,then all the exponents are even.  \n## Factorization\n\nPrime Numbers (Under 100):  \n2,3,5,7,11,13,17,19,23,2931,37,41,43,47,53,59,61,6771,73,79,83,89,97Prime FactorizationFor Example:72=2³×3²180=2²×3²×5525=3×5²×7  \n## Divison Method\n\n1.Divide the given number by the smallest divisible prime number  \n2.Divide the quotient by the smallest divisible prime number  \n3.Repeat the process until the quotient becomes a prime number.  \n4.Write the solution in standard form.  \n2  \n36  \n2  \n18  \n3  \n9  \n3  \n## The Total Number of Factors of a Natural Number\n\nThe Total Number of Factors of a Natural Numberr  \nFor a natural number n whose prime factorization is n=p1²¹·P2²·p₃³…Pe,we get the totalnumber of positive factors of this number by adding 1 to each exponent and then multiplyingthem together.  \nThe number of factors of n=(e₁+1)(e₂+1)…(ec+1).  \nThe number of factors of a perfect square is odd.  \nExample:  \n36=2².3²  \nThere are(2+1)(2+1)=9 factors of 36.  \nNotice the question is asking for prime factors or factors.  \n## The Greatest Common Factor and the Least Common Multiple:\n\nThe Greatest Common Factor and the Least Common Multiple:  \nTo find the greatest common factor(GCF)and least common multiple(LCM)of two or moreintegers,one way is using their prime factorization.To find the GCF,take the minimum exponent for each prime.To find the LCM,take the maximum exponent for each prime.  \nThink Academy  \nFor example,to find the GCF and LCM of 24,56,96:24=2³.3¹.7⁰56=2³.3°·7¹96=2⁵.3¹.7⁰  \nThe GCF is 2³.3⁰.7⁰=8.The LCM is 2⁵·3¹·7¹=672.  \n## Divisibility Rules\n\nDivisibility Rule:  \n| Divisor   | Divisibility Rules:  \u003Cbr>A k-digit integer N is divisible by   | Example   |\n| --- | --- | --- |\n| 2   | lif the last digit of N is divisible by 2(alleven  \u003Cbr>number is divisible by 2)   | \\   |\n| 5   | if the last digit of N is divisible by 5(the last  \u003Cbr>digit is 0 or5)   | \\   |\n| 4   | if the last 2 digits of N are divisible by 4   | \\   |\n| 25   | lif the last 2 digits of N are divisible by 25   | \\   |\n| 8   | if the last 3 digits of N are divisible by 8   | \\   |\n| 125   | if the last 3 digits of N are divisible by 125   | \\   |\n| 3   | if the sum of digits of N is divisible by 3   |  |\n| 9   | if the sum of digits of N is divisible by 9   | \\   |\n| 11   | if the alternating sum of digits of N is divisible  \u003Cbr>by 11   | 2728 is divisible by 11 because  \u003Cbr>(8+7)-(2+2)=11,which is  \u003Cbr>divisible by 11   |\n|  | if subtracting the first k-3 digits of N from the  \u003Cbr>remaining 3 digits gives a number that is  \u003Cbr>divisible by 11   | 263219 is divisible by 11  \u003Cbr>because 219-263=-44,  \u003Cbr>which is divisible by 11   |\n| 7   | if subtracting twice the last digit of N from the  \u003Cbr>remaining digits gives a number that is  \u003Cbr>divisible by 7   | 434 is divisible by 7because  \u003Cbr>43-2×4=35,which is  \u003Cbr>divisible by 7   |\n|  | if subtracting the first k-3 digits of N from the  \u003Cbr>remaining 3 digits gives a number that is  \u003Cbr>divisible by 7   | 16415 is divisible by 7 because  \u003Cbr>415-16=399,which is  \u003Cbr>divisible by 7   |\n| 13   | if subtracting the first k-3 digits of N from the  \u003Cbr>remaining 3 digits gives a number that is  \u003Cbr>divisible by 13   | 378703 is divisible by 13  \u003Cbr>because 703-378=325,  \u003Cbr>which is divisible by 13   |\n\nPythagorean TriplesPythagorean Theorema²+b²=c²Pythagorean Triples  \n(3,4,5)(5,12,13)(7,24,25)(8,15,17)(9,40,41)(12,35,37)  \n## Common Remainder Problems\n\n### Common Type of Remainder Problem:\n\nThe 1st type of problems:Different divisors have the same remainder in an equation.We canfind the dividend by finding the common multiple then add the r","cbCaiaDZlpAUr4N7","https://ap.wps.com/l/cbCaiaDZlpAUr4N7","pdf",1482685,7,"English","# Parity Rules\n# Perfect Squares & Properties\n## Common Primes\n## Common Perfect Squares\n# Factorization\n## Prime Numbers (Under 100)\n# Division Method\n# The Total Number of Factors of a Natural Number\n# The Greatest Common Factor and the Least Common Multiple\n# Divisibility Rules\n# Pythagorean Triples\n# Common Remainder Problems\n## Common Type of Remainder Problem:","[{\"question\":\"How do parity rules determine even/odd results for basic operations?\",\"answer\":\"The sheet states combinations such as even+even=even, even+odd=odd, and odd×odd=odd (with the other products listed similarly). Use these combinations to predict parity without full calculation.\"},{\"question\":\"What is the formula for the total number of positive factors from a prime factorization?\",\"answer\":\"For n with prime factorization n=p1^e1·p2^e2·…·pc^ec, the number of factors is (e1+1)(e2+1)…(ec+1). It also notes that a perfect square has an odd number of factors.\"},{\"question\":\"How can you find the GCF and LCM using prime factorization?\",\"answer\":\"For the GCF, take the minimum exponent of each prime across the numbers; for the LCM, take the maximum exponent of each prime. The sheet provides an example using 24, 56, and 96 to compute GCF=8 and LCM=672.\"}]","Number Theory Cheat Sheet | PDF",1791927116]