[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-142974-105":59,"doc-detail-142974-en":134},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":127,"head_meta":129,"extra_data":131,"updated_unix":133},105,"en","prime-numbers-factors-divisibility-tests-and-factor-trees","Prime Numbers - Factors, Divisibility Tests, and Factor Trees","","Prime numbers are defined as whole numbers greater than 1 whose only factors (divisors) are 1 and the number itself, equivalent to having exactly two divisors. The content distinguishes factors and divisors, explains divisibility through inverse operations, and gives practical divisibility tests for 2 through 9. Worked examples identify prime and composite numbers, list primes below specific limits, and show how to express composite numbers as products of prime factors using factor trees.",{"@graph":69,"@context":126},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/prime-numbers-factors-divisibility-tests-and-factor-trees/142974/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/prime-numbers-factors-divisibility-tests-and-factor-trees/142974.png","ImageObject",300,407,{"name":92,"@type":93},"Logic","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-28","2026-08-25",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118,122],{"name":109,"@type":110,"acceptedAnswer":111},"What is the definition of a prime number?","Question",{"text":112,"@type":113},"A prime number is a whole number other than 1 whose only factors (divisors) are 1 and itself. Equivalently, it has exactly two divisors.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do divisibility tests help determine divisibility?",{"text":117,"@type":113},"Divisibility tests provide quick rules based on the ending digits or the sum of digits. If the condition is met, the remainder is zero when dividing by the given divisor.",{"name":119,"@type":110,"acceptedAnswer":120},"Are factors and divisors the same thing?",{"text":121,"@type":113},"Yes. When c = a × b, both a and b are factors of c and also divisors of c, because each divides c with remainder zero.",{"name":123,"@type":110,"acceptedAnswer":124},"How can a composite number be expressed using prime factors?",{"text":125,"@type":113},"A composite number can be written as a product of prime numbers using a factor tree. The tree breaks the number into primes until all branches end at prime factors.","https://schema.org",{"og:url":83,"og:type":128,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":130,"canonical":83},"index,follow",{"doc_id":132,"site_id":62},142974,1787692109,{"code":4,"msg":5,"data":135},{"doc_id":132,"user_id":136,"nickname":92,"user_avatar":137,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":138,"file_id":139,"file_url":140,"file_type":141,"file_size":142,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":143,"language":144,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":145,"faqs":146,"seo_title":147,"seo_description":67,"update_tm":133,"read_time":36},1099513958762,"https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253","+99894 4905775 @westminstermath  \nIn the statement 3 􀀁 4 = 12, the number 12 is called the product, while 3 and 4 are called factors.  \n~~  ~~You~~ ~~Try~~ ~~It!~~  ~~  \nEXAMPLE 1 . Find all whole number factors of 18 .  \nSolution. We need to 􀀌nd all whole number pairs whose product equals 18 . The following pairs come to mind.  \n1 􀀁 18 = 18 and 2 􀀁 9 = 18 and 3 􀀁 6 = 18 .  \nHence, the factors of 18 are (in order) 1, 2, 3, 6, 9, and 18 .  \nFind all whole number factors of 21 .  \nAnswer: 1, 3, 7, and 21 .  \n ~~ ~~ 􀀃  \nDivisibility  \nIn Example 1, we saw 3 􀀁 6 = 18, making 3 and 6 factors of 18 . Because division is the inverse of multiplication, that is, divison by a number undoes the multiplication of that number, this immediately provides  \n18 􀀄 6 = 3 and 18 􀀄 3 = 6 .  \nThat is, 18 is divisible by 3 and 18 is divisible by 6 . When we say that 18 is divisible by 3, we mean that when 18 is divided by 3, there is a zero remainder.  \nDivisible. Let a and b be whole numbers. Then a is divisible by b if and only if the remainder is zero when a is divided by b. In this case, we say that “b is a divisor of a.”  \n~~  ~~You~~ ~~Try~~ ~~It!~~  ~~  \nEXAMPLE 2 . Find all whole number divisors of 18 .  \nSolution. In Example 1, we saw that 3 􀀁 6 = 18 . Therefore, 18 is divisible by both 3 and 6 (18 􀀄 3 = 6 and 18 􀀄 6 = 3) . Hence, when 18 is divided by 3 or 6, the remainder is zero. Therefore, 3 and 6 are divisors of 18 . Noting the other products in Example 1, the complete list of divisors of 18 is 1, 2, 3, 6, 9, and 18.  \nFind all whole number divisors of 21 .  \nAnswer: 1, 3, 7, and 21 .  \n ~~ ~~ 􀀃  \nExample 1 and Example 2 show that when working with whole numbers, the words factor and divisor are interchangeable.  \n+99894 4905775 @westminstermath  \nFactors and Divisors. If  \nc = a 􀀁 b,  \nthen a and b are called factors of c. Both a and b are also called divisors of c.  \nDivisibility Tests  \nThere are a number of very useful divisibility tests.  \nDivisible by 2 . If a whole number ends in 0, 2, 4, 6, or 8, then the number is  \ncalled an even number and is divisible by 2 . Examples of even numbers  \nare 238 and 1,246 (238 􀀄 2 = 119 and 1 , 246 􀀄 2 = 623) . A number that  \nis not even is called an odd number. Examples of odd numbers are 113  \nand 2,339 .  \nDivisible by 3 . If the sum of the digits of a whole number is divisible by 3, then the number itself is divisible by 3 . An example is 141 . The sum of the digits is 1 + 4 + 1 = 6, which is divisible by 3 . Therefore, 141 is also  \ndivisible by 3 (141 􀀄 3 = 47) .  \nDivisible by 4 . If the number represented by the last two digits of a whole  \nnumber is divisible by 4, then the number itself is divisible by 4 . An  \nexample is 11,524 . The last two digits represent 24, which is divisible by  \n4 (24 􀀄 4 = 6) . Therefore, 11,524 is divisible by 4 (11 , 524 􀀄 4 = 2 , 881) .  \nDivisible by 5 . If a whole number ends in a zero or a 5, then the number is  \ndivisible by 5 . Examples are 715 and 120 (715 􀀄 5 = 143 and 120 􀀄 5 = 24) .  \nDivisible by 6 . If a whole number is divisible by 2 and by 3, then it is divisible  \nby 6 . An example is 738 . First, 738 is even and divisible by 2 . Second,  \n7+3+8=18, which is divisible by 3 . Hence, 738 is divisible by 3 . Because  \n738 is divisible by both 2 and 3, it is divisible by 6 (738 􀀄 6 = 123) .  \nDivisible by 8 . If the number represented by the last three digits of a whole  \nnumber is divisible by 8, then the number itself is divisible by 8 . An  \nexample is 73,024 . The last three digits represent the number 24, which is  \ndivisible by 8 (24 􀀄 8 = 3) . Thus, 73,024 is also divisible by 8 (73 , 024 􀀄 8 =  \n9 , 128) .  \nDivisible by 9 . If the sum of the digits of a whole number is divisible by 9, then the number itself is divisible by 9 . An example is 117 . The sum of the digits is 1 + 1 + 7 = 9, which is divisible by 9 . Hence, 117 is divisible  \nby 9 (117 􀀄 9 = 13) .  \n+99894 4905775 @westminstermath  \nPrime Numbers  \nWe begin with the de􀀌niti","cbCaidx3FoajJaSh","https://ap.wps.com/l/cbCaidx3FoajJaSh","pdf",166137,16,"English","# Factors and Divisors\n# Divisibility\n## Divisible and divisor definition\n## Divisibility tests (2 to 9)\n# Prime Numbers\n## Definition and examples\n# Composite Numbers\n# Factor Trees","[{\"question\":\"What is the definition of a prime number?\",\"answer\":\"A prime number is a whole number other than 1 whose only factors (divisors) are 1 and itself. Equivalently, it has exactly two divisors.\"},{\"question\":\"How do divisibility tests help determine divisibility?\",\"answer\":\"Divisibility tests provide quick rules based on the ending digits or the sum of digits. If the condition is met, the remainder is zero when dividing by the given divisor.\"},{\"question\":\"Are factors and divisors the same thing?\",\"answer\":\"Yes. When c = a × b, both a and b are factors of c and also divisors of c, because each divides c with remainder zero.\"},{\"question\":\"How can a composite number be expressed using prime factors?\",\"answer\":\"A composite number can be written as a product of prime numbers using a factor tree. The tree breaks the number into primes until all branches end at prime factors.\"}]","Prime Numbers - Factors, Divisibility Tests, and Factor Trees | PDF"]