{"id":154072,"date":"2022-03-25T23:07:41","date_gmt":"2022-03-25T17:37:41","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/perfect-numbers-even-and-odd-perfect-numbers\/"},"modified":"2025-06-24T14:27:19","modified_gmt":"2025-06-24T08:57:19","slug":"perfect-numbers-even-and-odd-perfect-numbers","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/","title":{"rendered":"Perfect Numbers"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#What_are_Perfect_Numbers\" title=\"What are Perfect Numbers?\">What are Perfect Numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#What_are_Complete_Numbers\" title=\"What are Complete Numbers?\">What are Complete Numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#History_of_Perfect_Numbers\" title=\"History of Perfect Numbers\">History of Perfect Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#How_to_Find_a_Perfect_Number\" title=\"How to Find a Perfect Number?\">How to Find a Perfect Number?<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#Steps_to_Identify_Perfect_Numbers_Using_Expressions\" title=\"Steps to Identify Perfect Numbers Using Expressions\">Steps to Identify Perfect Numbers Using Expressions<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#Important_Notes_on_Perfect_Number\" title=\"Important Notes on Perfect Number\">Important Notes on Perfect Number<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#Solved_Example_on_Perfect_Numbers\" title=\"Solved Example on Perfect Numbers\">Solved Example on Perfect Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#Practice_Questions_on_Perfect_Number\" title=\"Practice Questions on Perfect Number\">Practice Questions on Perfect Number<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#Perfect_Number_FAQs\" title=\"Perfect Number FAQs\">Perfect Number FAQs<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#What_are_the_first_5_perfect_numbers\" title=\"What are the first 5 perfect numbers?\">What are the first 5 perfect numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#What_are_the_largest_perfect_numbers\" title=\"What are the largest perfect numbers?\">What are the largest perfect numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/perfect-numbers\/#Is_9_a_perfect_number\" title=\"Is 9 a perfect number?\">Is 9 a perfect number?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>A perfect number is a positive integer. In number theory, a perfect number is equal to the sum of its positive factors excluding the number itself. The smallest perfect number is 6. It is equal to the sum of 1,2 and 3.<\/p>\n<p>In this article, we will discuss Perfect numbers, their definition, the history of perfect numbers and more.<\/p>\n<p style=\"text-align: center;\"><strong>Also Check: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/curve\/\">Curve<\/a> <\/strong><\/p>\n<div class=\"card\" style=\"text-align: center;\"><a class=\"btn btn-primary\" href=\"https:\/\/infinitylearn.com\/one-stop-solutions-for-school-prep?utm_source=surge&amp;utm_medium=interlinking\"><br \/>\n<img loading=\"lazy\" class=\" lazyloaded\" style=\"width: 100%; border-radius: 10px;\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2024\/09\/one-stop-solution-for-school-exam.jpg\" alt=\"one-stop-solutions school exam\" width=\"1201\" height=\"636\" data-src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2024\/09\/one-stop-solution-for-school-exam.jpg\" \/><\/a><a class=\"btn btn-primary\" href=\"https:\/\/infinitylearn.com\/one-stop-solutions-for-school-prep?utm_source=surge&amp;utm_medium=interlinking\"><button><strong>One Stop Solutions for School Exam Preparation<\/strong><\/button><\/a><\/p>\n<div style=\"text-align: center;\">Boost your school preparation with our comprehensive guide for CBSE, ICSE, and State Board exams. Get all the resources you need in one place and excel in your academic journey. Discover the ultimate one-stop solution at Infinity Learn today!<\/div>\n<\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_are_Perfect_Numbers\"><\/span>What are Perfect Numbers?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A perfect integer is a positive integer that can be expressed as the sum of its positive factors except for the number itself. In other words, perfect numbers are the positive integers that are the sum of their divisors. In mathematics, 6 is the smallest perfect number. Number 6 can be written as the sum of its factors: 1, 2, and 3.<\/p>\n<p>Note: the factors used for the summation do not include the number itself.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_are_Complete_Numbers\"><\/span>What are Complete Numbers?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The sum of all the divisors of a number is equal to twice the number, such numbers are called complete numbers. One must note that all the perfect numbers are also complete numbers.<\/p>\n<p>For example: factors of 6: 1, 2, 3, and 6<\/p>\n<p>6 as perfect number: 1 + 2 + 3 = 6<\/p>\n<p>6 as a complete number: 1 + 2 + 3 + 6 = 12<\/p>\n<p style=\"text-align: center;\"><strong>Also Check: <\/strong><strong><a href=\"https:\/\/infinitylearn.com\/surge\/maths\/area-of-a-quadrilateral\/\">Area of Quadrilateral<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"History_of_Perfect_Numbers\"><\/span>History of Perfect Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The concept of perfect numbers might date back to ancient Egypt, where these numbers could have been encountered naturally. The Information regarding the discovery of the Perfect number is not known.<\/p>\n<p>This number got much attention from the Pythagoreans. It is a little of a mystery as to when Perfect Numbers were first discovered, by how they were studied, it is speculated that they may even have been of Egyptian origin, and may even have been known before (Burton, 1980).<\/p>\n<p>Although the ancient mathematicians knew of the existence of Perfect Numbers, it was the Greeks who took a keen interest in them, especially Pythagoras and his followers, for example, the scholars O\u2019Connor and Robertson (2004) write that every fifth number in the sequence of natural numbers, namely 6, 28, 496, and 8128 is perfect, that is it is equal to the sum of its proper divisors. Further, they imply that the Greeks did not know of the existence of them (O\u2019Connor and Robertson, 2004).<\/p>\n<p>The smallest perfect number is 6, which caught the attention of the Pythagoreans. They valued it not just for its mathematical properties but also for its mystical and numerological significance. Following 6, the next perfect number discovered was 28. These early observations marked the beginning of the exploration of perfect numbers in mathematics, blending early numerology with the roots of mathematical theory.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_to_Find_a_Perfect_Number\"><\/span>How to Find a Perfect Number?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To find a perfect number, you can use a technique used by Euclid. According to Euclid, an expression that can be a perfect number subject to a specific condition. According to his proposition, if 2n &#8211; 1 is a prime number then 2n-1(2n &#8211; 1) is a perfect number. Euclid said that 2n &#8211; 1 multiplied by 2n-1, is a perfect number. The term in the bracket that is (2n-1) is a prime number. In other words [2n-1 \u00d7 (2n-1) = perfect number], where (2n-1) is a prime number.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table table-bordered table-striped\" cellspacing=\"0\" cellpadding=\"5\">\n<tbody>\n<tr style=\"background-color: #89cff0; color: black;\">\n<td>n<\/td>\n<td>(2n-1)<\/td>\n<td>(2n-1)<\/td>\n<td>(2n-1) \u00d7 (2n &#8211; 1)<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>2<\/td>\n<td>3 (prime number)<\/td>\n<td>6 (perfect number)<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>4<\/td>\n<td>7 (prime number)<\/td>\n<td>28 (perfect number)<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>8<\/td>\n<td>15<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>16<\/td>\n<td>31 (prime number)<\/td>\n<td>496 (perfect number)<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>32<\/td>\n<td>63<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>64<\/td>\n<td>127 (prime number)<\/td>\n<td>8128 (perfect number)<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>128<\/td>\n<td>255<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>256<\/td>\n<td>511<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>512<\/td>\n<td>1023<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\"><strong>Also Check: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/average\/\">Average<\/a><\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Steps_to_Identify_Perfect_Numbers_Using_Expressions\"><\/span>Steps to Identify Perfect Numbers Using Expressions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Step 1:<\/strong> Start by substituting n=1 into the expressions 2n\u22121 and (2 n \u22121). Substituting n=1 into the first expression gives 21\u22121 = 1. Substituting n=1 into the second expression gives (21 \u2212 1) = 1.<\/p>\n<p><strong>Step 2: <\/strong>Continue substituting values of n=2,3,4, and so on, and record the results in a table.<\/p>\n<p><strong>Step 3:<\/strong> Observe the column containing (2n \u22121). If this number is prime, then the product of the two expressions, 2n\u22121 and (2n \u22121), will result in a perfect number.<\/p>\n<p><strong>Step 4:<\/strong> For example, when n=2, the first expression 22 &#8211; 1 gives 3, and the second expression (22 \u22121) gives 3, both of which are prime. Similarly, when n=3, the first expression gives 5, and the second expression gives 7.<\/p>\n<p><strong>Step 5:<\/strong> By examining the table, notice that the prime numbers under the column (2 n \u22121) correspond to perfect numbers when multiplied by their respective 2n\u22121 values. For example, prime numbers such as 3, 7, 31, and 127 lead to perfect numbers 6, 28, 496, and 8128, respectively. This demonstrates that when (2n \u22121) is prime, the product of 2n\u22121 and (2n\u22121) will always yield a perfect number.<\/p>\n<p>A few of the perfect numbers that are known throughout time are 6,28, etc. A list of a few more is provided below.<\/p>\n<ul>\n<li>6 (1 + 2 + 3)<\/li>\n<li>28 (1 + 2 + 4 + 7 + 14)<\/li>\n<li>496 (1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248)<\/li>\n<li>8128 (1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064)<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Important_Notes_on_Perfect_Number\"><\/span>Important Notes on Perfect Number<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>Perfect numbers are those which are both a positive integer and the sum of its proper divisors, excluding itself.<\/li>\n<li>6 is the smallest perfect number.<\/li>\n<li>All perfect numbers are even.<\/li>\n<li>Perfect numbers all end in 6 and 8 every other time.<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><strong>Also Check:<a href=\"https:\/\/infinitylearn.com\/surge\/maths\/circumference-of-a-circle\/\"> Circumstance of Circle<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solved_Example_on_Perfect_Numbers\"><\/span>Solved Example on Perfect Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Example 1: Check whether &#8217;28&#8217; is a perfect number.<\/strong><\/p>\n<p><strong>Solution: <\/strong>Factors of 28 = 1, 2, 4, 7, 14, 28<\/p>\n<p>The sum of factors excluding 28 is 1 + 2 + 4 + 7 + 14 = 28<\/p>\n<p>Here the sum of the factors of 28 excluding 28 is 28 itself.<\/p>\n<p>Therefore 28 is a perfect number.<\/p>\n<p><strong>Example 2: Check whether &#8216;6&#8217; is a perfect number.<\/strong><\/p>\n<p><strong>Solution: <\/strong>Factors of 6 = 1, 2, 3, 6<\/p>\n<p>The sum of factors excluding 6 is 1 + 2 + 3 = 6.<\/p>\n<p>Here the sum of the factors of 6 excluding is 6.<\/p>\n<p>Therefore 6 is a perfect number.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Practice_Questions_on_Perfect_Number\"><\/span>Practice Questions on Perfect Number<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>1. Check whether &#8217;62&#8217; is a perfect number.<\/p>\n<p>2. Check whether &#8216;986&#8217; is a perfect number.<\/p>\n<p>3. Check whether &#8216;697&#8217; is a perfect number.<\/p>\n<p>4. Check whether &#8217;23&#8217; is a perfect number.<\/p>\n<p>5. Check whether &#8216;245&#8217; is a perfect number.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Perfect_Number_FAQs\"><\/span>Perfect Number FAQs<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_the_first_5_perfect_numbers\"><\/span>What are the first 5 perfect numbers?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe first 5 perfect numbers are 6, 28, 496, 8128, and 33550336.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_the_largest_perfect_numbers\"><\/span>What are the largest perfect numbers?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe largest perfect number is unknown.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Is_9_a_perfect_number\"><\/span>Is 9 a perfect number?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tNo, nine is not a perfect number.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are the first 5 perfect numbers?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The first 5 perfect numbers are 6, 28, 496, 8128, and 33550336.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are the largest perfect numbers?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The largest perfect number is unknown.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Is 9 a perfect number?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"No, nine is not a perfect number.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>A perfect number is a positive integer. In number theory, a perfect number is equal to the sum of its [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Perfect Numbers | Even and Odd Perfect Numbers","_yoast_wpseo_title":"Perfect Numbers | Even and Odd Perfect Numbers","_yoast_wpseo_metadesc":"A perfect number is a positive integer that equals the sum of its proper divisors, that is, positive divisors excluding the number itself.","custom_permalink":"maths\/perfect-numbers\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Perfect Numbers | Even and Odd Perfect Numbers<\/title>\n<meta name=\"description\" content=\"A perfect number is a positive integer that equals the sum of its proper divisors, that is, positive divisors excluding the number 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