Canonical representation of a number:<\/h5>\n
Every integer can be expressed as the product of primes and this factoring of the integer into primes is unique except the order of primes.<\/p>\n
So, any integer n can be expressed as, n = p1<\/sub>k1<\/sup>xp2<\/sub>k2<\/sup>x \u2026x pm<\/sub>km<\/sup>, where p1<\/sub>,p2<\/sub>,…,pm<\/sub> are primes where 1<p1<\/sub><p2<\/sub><\u2026<pk<\/sub> and k1<\/sub>,k2<\/sub>,\u2026,km<\/sub> are positive integers.<\/p>\n If we take a number 6, we can say that there are only 4 divisors for 6 which are 1,2,3,6. Similarly if we take a number 24 we say that there are 8 divisors namely 1,2,3,4,6,8,12,24. But what if we have to find the number of divisors of a big number such as 2000. There is a formula for finding the number of divisors of a given number.<\/p>\n If n = p1<\/sub>k1<\/sup>xp2<\/sub>k2<\/sup>x \u2026x pm<\/sub>km<\/sup> is the canonical representation of a number n then the number of positive divisors of n is given by (1+k1<\/sub>)(1+k2<\/sub>)\u2026(1+km<\/sub>).<\/p>\n Example:<\/strong><\/p>\n 2000 = 24<\/sup> x 53<\/sup><\/p>\n Number of positive divisors of 2000 = (1+4)(1+3) = 5×4 = 20.<\/p>\n As we discussed earlier, when it is difficult to find the number of divisors, it is obviously more difficult to find the sum of divisors. There is a formula for finding the sum of divisors of a given number.<\/p>\n If n = p1<\/sub>k1<\/sup>xp2<\/sub>k2<\/sup>x \u2026x pm<\/sub>km<\/sup> is the canonical representation of a number n then the sum of positive divisors of n is given by<\/p>\n Example:<\/strong><\/p>\n 2000 = 24<\/sup> x 53<\/sup><\/p>\n Sum of positive divisors of 2000 =[25<\/sup>-1\/2-1][54<\/sup>-1\/5-1] = [31\/1] x [624\/4] = 31 x 156 = 4836. Prime and composite numbers: An integer is called prime if it is divisible only by 1 and itself. i.e. a prime number cannot be divisible by any other number than 1 and itself. A prime number has exactly 2 positive divisors. A number which has more than 2 positive divisors is said to be a […]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[225,347,536,1217,1385,1533,1774],"class_list":["post-3417","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-canonical","tag-composite","tag-divisors","tag-number","tag-prime","tag-representation","tag-sum"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3417","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=3417"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3417\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=3417"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=3417"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=3417"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Number of all positive divisors of a number:<\/h5>\n
Sum of all positive divisors of a number:<\/h5>\n
![\"\"](\"https:\/\/SchoolTutoring.com\/wp-content\/uploads\/sites\/2\/2012\/08\/canonical.jpg\" "\\"canonical\\"")
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