[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-347412-en":3,"doc-seo-347412-105":31,"detail-sidebar-cat-0-en-105":102},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30,"preview_page_count":4},347412,1099523885336,"Taylor Morgan","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",4,"Exam","Important Concepts for CAT - Number Theory","Study notes covering key CAT mathematics foundations in number theory and factor-related concepts. The content defines integers, whole and natural numbers, factors, factor foundation rule, multiples, prime numbers, prime factorization, greatest common factor (GCF), and least common multiple (LCM). It also summarizes parity rules for odd/even and introduces divisibility patterns for products and sums of consecutive integers, supported by multiple “mantra” shortcuts and structured divisibility tests.","TGSINCE 2007LEADERS IN TEST-PREPIMPORTANT CONCEPTSFOR CAJ  \nTathaGat  \n# Important Concepts for CAT\n\n# Number Theory\n\n## ●Integers\n\nNumbers,such as-1,0,1,2,and 3,that have no fractional part.Integers include the  \ncounting numbers(1,2,3,…),their negative counterparts(-1,-2,-3,….),and 0.  \n## ●Whole &Natural Numbers\n\nThe terms from 0,1,2,3,….are known as Whole numbers.Natural numbers do not include  \n0.  \n## ●Factors\n\nPositive integers that divide evenly into an integer.Factors are equal to or smaller than theinteger in question.12 is a factor of 12,as are 1,2,3,4,and 6.  \n## ●Factor Foundation Rule\n\nIf a is a factor of b,and b is a factor of c,then a is also a factor of c.For example,5 is afactor of 25 and 25 is a factor of 625.Therefore,5 is also a factor of 625.  \n## ●Multiples\n\nMultiples are integers formed by multiplying some integer by any other integer.Forexample,6 is a multiple of 3(2×3),as are 12(4×3),18(6×3),etc.In addition 3 isalso a multiple of itself i.e.3(1×3).Think of multiples as equal to or larger than theinteger in question  \n## ●Prime Numbers\n\nA positive integer with exactly two factors:1 and itself.The number 1 does not qualify asprime because it has only one factor,not two.The number 2 is the smallest prime number;it is also the only even prime number.The numbers 2,3,5,7,11,13 etc.are prime.  \n## ●Prime Factorization\n\nPrime factorization is a way to express any number as a product of prime numbers.Forexample,the prime factorization of 30 is 2×3×5.Prime factorization is useful inanswering questions about divisibility.  \n## ●Greatest Common Factor\n\nGreatest Common FACTOR refers to the largest factor of two(or more)integers.Factors willbe equal to or smaller than the starting integers.The GCF of 12 and 30 is 6 because 6 is thelargest number that goes into both 12 and 30.  \n## ●Least Common Multiple(LCM)\n\nLeast Common Multiple refers to the smallest multiple of two(or more)integers.Multipleswill be equal to or larger than the starting integers.The LCM of 6 and 15 is 30 because 30 isthe smallest number that both 6 and 15 go into.  \n## ●Odd &Even Numbers\n\nAny number divisible by 2 is even and not divisible by 2 is odd.  \n## ●Product of n consecutive integers and divisibility\n\nThe product of n consecutive integers is always divisible by n!Given 5×6×7×8,we have n =4 consecutive integers.The product of 5×6×7×8(=1680),therefore,is divisible by 4!=5×6×7×8=24.  \n·Sum of n consecutive integers and divisibility  \nThere are two cases,depending upon whether n is odd or even:  \nIf n is odd,the sum of the integers is always divisible by n.Given 5+6+7,we have n =3consecutive integers.The sum of 5+6+7(=18),therefore,is divisible by 3.  \nIf n is even,the sum of the integers is never divisible by n.Given 5+6+7+8,we have n =4consecutive integers.The sum of 5+6+7+8(=26),therefore,is not divisible by 4.  \n·MANTRA:Square of any natural number can be written in the form of 3n or 3n+1.Also,square of any natural number can be written in the form of 4n or 4n+1.  \n·MANTRA:Square of a natural number can only end in 0,1,4,5,6 or 9.Second last digitof a square of a natural number is always even except when last digit is 6.If the last digit is  \n5,second last digit has to be 2.  \n●MANTRA:Any prime number greater than 3 can be written as 6k±1.  \n●MANTRA:The fifth power of any number has the same units place digit as the numberitself.  \n·MANTRA:Cube on any natural number can be written in the form of 7n,7n +1 or 7n-1  \nAlso,cubw of any natural number can be written in the form 9n,9n-1 or 9n +1.  \n●MANTRA:If two numbers a and b are given,and their LCM and HCF are L and Hrespectively,then L×H=a×b.  \n## ●MANTRA:\n\n●MANTRA:  \nNote:Fractions should be in the lowest form.  \n·MANTRA The least number leaving remainder'r'in each case when divided by'x′,`y'and`z′=(LCM of x,y,z)+r  \n●MANTRA The series of such numbers will be(LCM of x,y,z)×n+r  \n# Divisibilty Rules\n\n·Divisibility Rules for the numbers of the form 10”-1.  \nFor checking divisibility by 'p’,w","cbCaigiC47C0NLIc","https://ap.wps.com/l/cbCaigiC47C0NLIc","pdf",2717004,17,1,30,"English","en",105,"# Important Concepts for CAT\n# Number Theory\n## ●Integers\n## ●Whole & Natural Numbers\n## ●Factors\n## ●Factor Foundation Rule\n## ●Multiples\n## ●Prime Numbers\n## ●Prime Factorization\n## ●Greatest Common Factor\n## ●Least Common Multiple (LCM)\n## ●Odd & Even Numbers\n## ●Product of n consecutive integers and divisibility\n## ●Sum of n consecutive integers and divisibility\n## ●MANTRA\n# Divisibility Rules\n## ●Divisibility Rules for 10^n-1 forms\n## ●Divisibility Rules for 10^n+1 forms\n# FACTOR THEORY","[{\"question\":\"How are integers, whole numbers, and natural numbers distinguished?\",\"answer\":\"Integers include all numbers with no fractional part, including negatives and 0. Whole numbers are 0 and the counting numbers, while natural numbers exclude 0.\"},{\"question\":\"What do GCF and LCM mean and how are they related to factors and multiples?\",\"answer\":\"Greatest Common Factor (GCF) is the largest factor common to two or more integers. Least Common Multiple (LCM) is the smallest multiple shared by two or more integers.\"},{\"question\":\"Which divisibility tests are provided for numbers of the form 10^n-1 and 10^n+1?\",\"answer\":\"For 10^n-1, divisibility is checked using sums of blocks of size n (from the least significant digits). For 10^n+1, divisibility is checked using alternating sums of blocks of size n (from the least significant digits).\"},{\"question\":\"What “mantra” rules help with prime numbers and powers?\",\"answer\":\"The notes state that any prime number greater than 3 can be written as 6k±1, and that the fifth power shares the same units digit as the original number. They also provide forms for cubes and squares based on modular patterns.\"}]","Important Concepts for CAT - Number Theory | PDF",1790066816,76,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":96,"head_meta":98,"extra_data":100,"updated_unix":101},"important-concepts-for-cat-number-theory","",{"@graph":37,"@context":95},[38,54,74],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/exam/",3,{"item":53,"name":13,"@type":44,"position":11},"https://docshare.wps.com/document/important-concepts-for-cat-number-theory/347412/",{"url":53,"name":13,"@type":55,"image":56,"author":61,"headline":13,"publisher":63,"fileFormat":66,"inLanguage":24,"description":14,"dateModified":67,"datePublished":68,"encodingFormat":66,"isAccessibleForFree":69,"interactionStatistic":70},"DigitalDocument",{"url":57,"@type":58,"width":59,"height":60},"https://docshare.wps.com/thumbnails/important-concepts-for-cat-number-theory/347412.png","ImageObject",300,407,{"name":9,"@type":62},"Person",{"url":42,"name":64,"@type":65},"DocShare","Organization","application/pdf","2026-11-07","2026-09-22",true,{"@type":71,"interactionType":72,"userInteractionCount":20},"InteractionCounter",{"@type":73},"ViewAction",{"@type":75,"mainEntity":76},"FAQPage",[77,83,87,91],{"name":78,"@type":79,"acceptedAnswer":80},"How are integers, whole numbers, and natural numbers distinguished?","Question",{"text":81,"@type":82},"Integers include all numbers with no fractional part, including negatives and 0. Whole numbers are 0 and the counting numbers, while natural numbers exclude 0.","Answer",{"name":84,"@type":79,"acceptedAnswer":85},"What do GCF and LCM mean and how are they related to factors and multiples?",{"text":86,"@type":82},"Greatest Common Factor (GCF) is the largest factor common to two or more integers. Least Common Multiple (LCM) is the smallest multiple shared by two or more integers.",{"name":88,"@type":79,"acceptedAnswer":89},"Which divisibility tests are provided for numbers of the form 10^n-1 and 10^n+1?",{"text":90,"@type":82},"For 10^n-1, divisibility is checked using sums of blocks of size n (from the least significant digits). For 10^n+1, divisibility is checked using alternating sums of blocks of size n (from the least significant digits).",{"name":92,"@type":79,"acceptedAnswer":93},"What “mantra” rules help with prime numbers and powers?",{"text":94,"@type":82},"The notes state that any prime number greater than 3 can be written as 6k±1, and that the fifth power shares the same units digit as the original number. They also provide forms for cubes and squares based on modular patterns.","https://schema.org",{"og:url":53,"og:type":97,"og:title":13,"og:site_name":64,"og:description":14},"article",{"robots":99,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},1790241092,{"code":4,"msg":5,"data":103},[104,108,112,115,120,125,130,134,139,142,146],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":105,"show_sort_weight":106,"slug":107},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":113,"slug":114},70,"exam",{"id":116,"doc_module":4,"doc_module_name":47,"category_name":117,"show_sort_weight":118,"slug":119},5,"Comic",60,"comic",{"id":121,"doc_module":4,"doc_module_name":47,"category_name":122,"show_sort_weight":123,"slug":124},6,"Technology",50,"technology",{"id":126,"doc_module":4,"doc_module_name":47,"category_name":127,"show_sort_weight":128,"slug":129},7,"Healthcare",40,"healthcare",{"id":131,"doc_module":4,"doc_module_name":47,"category_name":132,"show_sort_weight":22,"slug":133},8,"Research & Report","research-report",{"id":135,"doc_module":4,"doc_module_name":47,"category_name":136,"show_sort_weight":137,"slug":138},9,"Religion & Spirituality",20,"religion-spirituality",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":140,"show_sort_weight":137,"slug":141},"World Cup","world-cup",{"id":143,"doc_module":4,"doc_module_name":47,"category_name":144,"show_sort_weight":143,"slug":145},10,"Lifestyle","lifestyle",{"id":147,"doc_module":4,"doc_module_name":47,"category_name":148,"show_sort_weight":116,"slug":149},19,"General","general"]