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It explains formulas for sums of natural numbers, even/odd series, squares, cubes, and includes rules for last digits and digit-sum tests for numbers like 3, 7, 9, and 11. It further introduces remainders, modulus interpretation, and remainder theorems including Wilson’s and Euler’s totient approach with examples.",{"@graph":69,"@context":123},[70,84,106],{"@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/useful-formulae-concepts-for-quantitative-aptitude-an-iquanta-presentation/347410/",{"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/useful-formulae-concepts-for-quantitative-aptitude-an-iquanta-presentation/347410.png","ImageObject",300,407,{"name":92,"@type":93},"Riley West","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-27","2026-09-22",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What are the key divisibility rules for 2, 4, and 8?","Question",{"text":113,"@type":114},"A number is divisible by 2 if its last digit is even. It is divisible by 4 if the last two digits are 00 or divisible by 4, and divisible by 8 if the last three digits are 000 or divisible by 8.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How can you determine divisibility by 3, 9, 7, and 11?",{"text":118,"@type":114},"Divisible by 3 when the sum of digits is divisible by 3; divisible by 9 when the digit sum is divisible by 9. Divisible by 7 when the difference between twice the unit digit and the remaining part is a multiple of 7 (or 0). Divisible by 11 when the difference between sums of alternate digits is 0 or divisible by 11.",{"name":120,"@type":111,"acceptedAnswer":121},"What do remainder and modulus mean in the remainder theorem context?",{"text":122,"@type":114},"Remainder is what remains after dividing a by a divisor, expressed as a = divisor × quotient + remainder. Modulus means remainder; thus a mod b equals the remainder obtained on dividing a by b.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},347410,1790192105,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},1099523885074,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","iQuantaBuilding Learning Netwom423000二T℃C=2C喝mmath盛手CAT4=TC  \n码  \n# USEFUL FORMULAE&CONCEPTS FOR\n\nQUANTITATIVE APTITUDE  \n# An iQuanta Presentation\n\n# INDEX\n\n1  \nNumber  \nSystem  \nwww.iQuanta.in  \nNUMBER SYSTEM  \n## Important points to remember:\n\n1.The product of 'n’consecutive natural numbers is always divisible by n!  \n2.Square of any natural number can be written in the form of 3n or 3n+1.  \n3.Square of a natural number can only end in 0,1,4,5,6 or 9.Second lastdigit of a square of a natural number is always even except when last digit is  \n### 6.If the last digit is 5,second last digit has to be 2.\n\n4.Any prime number greater than 3 can be written as 6k+1 or 6k-1.  \n5.Any two-digit number 'mn'can effectively be written as 10m+n and a  \nthree-digit number Imn'can effectively be written as 1001+10m+n.  \n6.Sum of 1st n natural number,1+2+3+….+n =∑n =n(n+1)/2  \n7.Sum of 1st n positive even numbers,2+4+6-------+2n =n(n+1)  \n8.Sum of 1st n positive odd numbers,1+3+5+------(2n-1)=n²  \n9.Sum of squares of 1st n natural numbers:12+22+32+-----n²=n(n+1)(2n+1)/6  \n10.Sum of cubes of 1st n natural numbers:13+2³+33+--------n³=(∑n)²=[n(n+1)/2]2  \n11.0 and 1 are neither composite nor prime.  \n12.There are 25 prime numbers less than 100.  \n13.There are 15 prime numbers less than 50.  \nAdarsh Khandelwal,CAT 99.93%ler fromiQuanta shares gratitude  \nAdarsh Khandelwal  \n21 December at 22:44·母  \n\"Ups and downs in life are very important to keep us goingbecause a straight line even in an ECGmeans we are not alive.\"-Ratan Tata.A quote very close to my heart.  \nFrom scoring 99.90%ile in JEE Main to being diagnosed with a chronic health problem and notbeing able to take JEE Advanced and being ill for about 4 years,due to which I had to take up adistance-learnng course from IGNOU and graduated as BA(Pol Science)in 2019 along withbattling my disease up till September 2021 and finally winning.But I always knew that \"Apna timeaayega\".Fast forward to June 2022,though not able to get into IITs,I started paving way for myultimate goal of getting into IIMs and took a challenge from my brother to crack CAT 2022 with apercentile greater than that of JEE Main's (so yay!).  \nI started looking for the best coaching out there and stumbled upon a Quora article that directedme towards iQuanta.I enrolled in CAT 2022 Batch 2,at this point 1 must mention that at first I wasa bit skeptical about iQuanta and thought that how can any CAT coaching has a fee which is 25-30k less than some of the big names,but nevertheless I decided to buy it and I must say I wouldhave regretted not joining iQuanta.The dual pedagogy,24x7 doubt solving and peer-to-peerlearning helped me grow a lot in a very short time.I scored 99.93%ile and for that I would like tothank Jeet Singh,Abhishek Leela Pandey and Sajjan Barnwal for always being there and taking usthrough a truly awesome journey e.  \nP.S.-Jeet Singh Sir,ab GDPI ki naao bhi paar lagwa dijiye 6??  \n\n| Section   |  | Section   |  | Section   |  | Total   |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Verbal Ability &  \u003Cbr>Reading Comprehension   |  | Data Interpretation &  \u003Cbr>Logical Reasoning   |  | Quantitative Ability   |  |  |  |\n| Scaled  \u003Cbr>Score   | Percentile   | Scaled  \u003Cbr>Score   | Percentile   | Sealed  \u003Cbr>Score   | Percentile   | Overall  \u003Cbr>Scaled Score   | Overall  \u003Cbr>Percentile   |\n| 16.46   | 77.59   | 48.31   | 99.99   | 49.38   | 99.95   | 114.15   | 99.93   |\n\nFor iQuanta CAT Course,Visit www.iQuanta.in  \n## Divisibility rules\n\n2-when last digit of the number is 0,2,4,6,8.  \n4-when last two digits of the number is 00 or divisible by 4.  \n8-when the last three digits of the number is 000 or divisible by 8.  \n5-when last digit is O or 5.  \n2n-when last n digits of the numbers are divisible by 2n.  \n5n-when last n digits of the number are divisible by 5n.  \n### Divisibility rule for 3 and 9\n\n3-Sum of the digits should be divisible by 3.9-Sum of digits should be divisible by 9.  \n### Divisibility rule fo","cbCaij67bVzaYGxG","https://ap.wps.com/l/cbCaij67bVzaYGxG","pdf",6842132,59,"English","# INDEX\n## Important points to remember\n## Divisibility rules\n## Divisibility rule for 3 and 9\n## Divisibility rule for 7 and 11\n## Divisibility rules of 6,12,14,15,18,etc.\n## Properties of factors and co-prime numbers\n## Remainders and Remainder Theorems\n### 1.Wilson's Remainder Theorem\n### 2.Euler's Totient Theorem","[{\"question\":\"What are the key divisibility rules for 2, 4, and 8?\",\"answer\":\"A number is divisible by 2 if its last digit is even. It is divisible by 4 if the last two digits are 00 or divisible by 4, and divisible by 8 if the last three digits are 000 or divisible by 8.\"},{\"question\":\"How can you determine divisibility by 3, 9, 7, and 11?\",\"answer\":\"Divisible by 3 when the sum of digits is divisible by 3; divisible by 9 when the digit sum is divisible by 9. Divisible by 7 when the difference between twice the unit digit and the remaining part is a multiple of 7 (or 0). Divisible by 11 when the difference between sums of alternate digits is 0 or divisible by 11.\"},{\"question\":\"What do remainder and modulus mean in the remainder theorem context?\",\"answer\":\"Remainder is what remains after dividing a by a divisor, expressed as a = divisor × quotient + remainder. Modulus means remainder; thus a mod b equals the remainder obtained on dividing a by b.\"}]","USEFUL FORMULAE & CONCEPTS FOR QUANTITATIVE APTITUDE - An iQuanta Presentation | PDF",1790066815,149]