{"id":27535,"date":"2023-04-14T06:57:30","date_gmt":"2023-04-14T06:57:30","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=27535"},"modified":"2023-11-15T07:14:43","modified_gmt":"2023-11-15T07:14:43","slug":"what-are-perfect-numbers-definition-list-formula-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers","title":{"rendered":"What Are Perfect Numbers? Definition, List, Formula, Examples"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-ace5356f-6b4f-406b-bcb3-3ef401c9416d\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers#0-what-is-a-perfect-number>What Is a Perfect Number?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers#3-how-to-find-a-perfect-number>How to Find a Perfect Number<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers#5-perfect-numbers-from-1-to-100>Perfect Numbers From 1 to 100<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers#8-solved-examples-on-perfect-numbers>Solved Examples on Perfect Numbers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers#9-practice-problems-on-perfect-numbers>Practice Problems on Perfect Numbers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers#10-frequently-asked-questions-on-perfect-numbers>Frequently Asked Questions on Perfect Numbers<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-is-a-perfect-number\">What Is a Perfect Number?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Perfect numbers in math are the numbers (positive integers) that can be expressed as the sum of their factors (excluding the number itself)<\/strong>.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Factors of a number are the numbers that divide a given number exactly without leaving a remainder. A factor of a number is always less than or equal to the number.<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_make_10_pt.png\" alt=\"Add 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-2-digit-and-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_2d_1d_match_pt.png\" alt=\"Add 2-Digit and 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 2-Digit and 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-2-digit-numbers-by-regrouping\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_regrp_2d_2d_vertical_pt.png\" alt=\"Add 2-Digit Numbers By Regrouping Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 2-Digit Numbers By Regrouping Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_3_no_pt.png\" alt=\"Add 3 Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-in-any-order\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_shifty_bridges_4_gm.png\" alt=\"Add 3 Numbers in Any Order Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers in Any Order Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-using-groups-of-objects\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_using_model_pt.png\" alt=\"Add 3 Numbers Using Groups of Objects Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers Using Groups of Objects Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-numbers-using-model\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_add_3_no_using_models_pt.png\" alt=\"Add 3 Numbers using Model Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3 Numbers using Model Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-digit-and-1-digit-numbers-and-match\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_3d_1d_match_g3_pt.png\" alt=\"Add 3-Digit and 1-Digit Numbers and Match Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3-Digit and 1-Digit Numbers and Match Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-digit-and-1-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_3d_1d_vertical_pt.png\" alt=\"Add 3-Digit and 1-Digit Numbers Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3-Digit and 1-Digit Numbers Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-3-digit-and-1-digit-numbers-with-regrouping\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_pv_add_regrp_3d_1d_vertical_pt.png\" alt=\"Add 3-Digit and 1-Digit Numbers with Regrouping Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add 3-Digit and 1-Digit Numbers with Regrouping Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"1-perfect-numbers-definition\">Perfect Numbers Definition<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">A perfect number is defined as a positive integer that can be expressed as the sum of its proper factors (factors except for the number itself).&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Perfect number examples<strong>: 6, 28, 496<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The factors of 6 are 1, 2, 3 and 6.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">We can write $6 = 1 + 2 + 3$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The smallest perfect number is 6.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"315\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/28-as-a-perfect-number-expressed-as-sum-of-its-proper-factors.png\" alt=\"28 as a perfect number expressed as sum of its proper factors\" class=\"wp-image-35675\" title=\"28 as a perfect number expressed as sum of its proper factors\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/28-as-a-perfect-number-expressed-as-sum-of-its-proper-factors.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/28-as-a-perfect-number-expressed-as-sum-of-its-proper-factors-300x152.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>When the sum of all the divisors of a number is equal to twice the number, the number is known as complete number<\/strong>. All the perfect numbers are also complete numbers.<\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-subtract-ones-2-digit-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-subtract-ones-2-digit-numbers.jpeg\" alt=\"Add & Subtract Ones & 2-Digit Numbers Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-find-the-sum\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-find-the-sum.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers and Find the Sum\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-match-the-sum\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-and-match-the-sum.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers and Match the Sum\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-tables\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-tables.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Addition Tables\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-wheel\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-addition-wheel.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Addition Wheel\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-number-charts\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-number-charts.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Number Charts\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-place-value\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-and-3-digit-numbers-using-place-value.jpeg\" alt=\"Add 1 Digit and 3 Digit Numbers Using Place Value\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-numbers-and-teen-numbers-horizontal-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-numbers-and-teen-numbers-horizontal-addition.jpeg\" alt=\"Add 1-Digit Numbers and Teen Numbers: Horizontal Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-1-digit-numbers-and-teen-numbers-missing-digits\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-1-digit-numbers-and-teen-numbers-missing-digits.jpeg\" alt=\"Add 1-Digit Numbers and Teen Numbers: Missing Digits Worksheet\">\r\n\t    <\/div>\r\n\t\t<div 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class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-theorem-of-even-perfect-numbers\">Theorem of Even Perfect Numbers<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">An even natural number N is perfect if and only if it is of the form $N = 2^{n-1}(2^n\\;\u2212\\;1)$, where $2^n\\;\u2212\\;1$ is a prime number.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Example:<\/strong> Put $n = 2\\; in(2^n\\;\u2212\\;1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$2^n\\;\u2212\\;1 = 2^2\\;\u2212\\;1 = 3$ \u2026a prime number<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, $2^{n-1}(2^n\\;\u2212\\;1) = 2^{2-1}\\;(3) = 6$ \u2026a perfect number<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-how-to-find-a-perfect-number\">How to Find a Perfect Number<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">According to Euclid, there is an expression that can be used to check if a number is a perfect number. Around 2000 years ago, Euclid showed that any even perfect number can be represented by,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">If $(2^n\\;-\\;1)$ is a prime number, then $2^{n-1}(2^n\\;-\\;1)$&nbsp; is a perfect number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">In other words, $2^{n-1}(2^n\\;-\\;1)$ is a perfect number, if (2n-1) is a prime number.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let\u2019s find a value of \u201cn\u201d for which&nbsp; $(2^n\\;-\\;1)$ is prime. All such numbers will help us form a list of perfect numbers.<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>n<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\">$2^{n-1}$<\/th><th class=\"has-text-align-center\" data-align=\"center\">$(2^n\\;-\\; 1)$<\/th><th class=\"has-text-align-center\" data-align=\"center\">$2^{n-1} \\times (2^n\\;-\\; 1)$<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>2<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>2<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>3 (prime number)<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>6 (perfect number)<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>3<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>4<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>7 (prime number)<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>28 (perfect number)<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">15 (not a prime number)<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>5<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>16<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>31 (prime number)<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>496 (perfect number)<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">32<\/td><td class=\"has-text-align-center\" data-align=\"center\">63 (not a prime number)<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>7<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>64<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>127 (prime number)<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>8128&nbsp; (perfect number)<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">128<\/td><td class=\"has-text-align-center\" data-align=\"center\">255 (not a prime number)<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">256<\/td><td class=\"has-text-align-center\" data-align=\"center\">511 (not a prime number)<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">10<\/td><td class=\"has-text-align-center\" data-align=\"center\">512<\/td><td class=\"has-text-align-center\" data-align=\"center\">1023 (not a prime number)<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-perfect-numbers-chart-\">Perfect Numbers Chart&nbsp;<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">A few perfect numbers are 6, 28, 496 and 8128. Let us observe their divisors and their sum through the table given below. Their sum results in the number itself.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><thead><tr><th><strong>Perfect Number<\/strong><\/th><th><strong>Sum of Divisors<\/strong><\/th><\/tr><\/thead><tbody><tr><td><strong>6<\/strong><\/td><td>$1 + 2 + 3 = 6$<\/td><\/tr><tr><td><strong>28<\/strong><\/td><td>$1 + 2 + 4 + 7 + 14 = 28$<\/td><\/tr><tr><td><strong>496<\/strong><\/td><td>$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$<\/td><\/tr><tr><td><strong>8128<\/strong><\/td><td>$1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-perfect-numbers-from-1-to-100\"><br>Perfect Numbers From 1 to 100<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">There are only two perfect numbers between 1 to 100, which are 6 and 28.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-history-of-perfect-numbers\">History of Perfect Numbers<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The history of perfect numbers and its study goes back to the ancient Egyptians, Greeks, and also the followers of Pythagoras.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The smallest perfect number found out was 6, which encouraged people to study the properties of the perfect numbers. The early definition of perfect numbers was given as: <strong>a perfect number n is a positive integer, which is equal to the sum of its factors, excluding n itself<\/strong>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-facts-about-perfect-numbers\">Facts about Perfect Numbers<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">For any number, the number itself is the greatest factor. A proper factor of a number is any factor of the number except the number itself.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">There\u2019s a quote from Philo of Alexandria, a Jewish Philosopher from Egypt (born 25 BCE) that says: <strong><em>The world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect.<\/em><\/strong><\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">A Mersenne Prime is a prime number that is one less than a power of two. That means it is a prime number of the form $M_n = 2^n \\;\u2212\\; 1$ for some integer n. Examples: 3, 7, 31<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-solved-examples-on-perfect-numbers\">Solved Examples on Perfect Numbers<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>1. Is 6 a perfect number?&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Yes. $6 = 1 + 2 + 3$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">where 1, 2 and 3 are the proper factors of 6.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>2. Check whether 28 is a perfect number or not by finding the sum of its factors.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The factors of 28 are 1, 2, 4, 7, 14 and 28.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Proper factors of 28 are 1, 2, 4, 7, and 14.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Sum of proper factors $= 1 + 2 + 4 + 7 + 14 = 28$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, 28 is a perfect number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>3. Is 15 a perfect number?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Factors of 15 are 1, 3, 5 and 15.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$1 + 3 + 5 = 9 \\neq 15$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, 15 is not a perfect number.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>4. If we substitute <\/strong>$n = 11$ in $2^n\\;-\\;1$<strong>, will it give us a perfect number?<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$n = 11$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">If $(2^n\\;-\\;1)$ is a prime number, then $2^n\\;-\\;1(2^n\\;-\\;1)$&nbsp; is a perfect number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">So, $2^11\\;-\\;1 = 1023$ which is not a prime number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">So, it will not give us a perfect number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>5. Check whether 39 is a perfect number by finding the sum of its factors.<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Factors of $39 = 1, 3, 13$ and 39.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Proper factors of $39 = 1, 3, 13$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$1 + 3 + 13 = 17 \\neq 39$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, 39 is not a perfect number.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-practice-problems-on-perfect-numbers\">Practice Problems on Perfect Numbers<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">What Are Perfect Numbers? Definition, List, Formula, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Which of the following is a perfect number?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">28<\/div><div class=\"spq_answer_block\" data-value=\"1\">36<\/div><div class=\"spq_answer_block\" data-value=\"2\">49<\/div><div class=\"spq_answer_block\" data-value=\"3\">125<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 28<br\/>Only 28 can be written as the sum of its proper factors<br>\r\n$28 = 1 + 2 + 4 + 7 + 14$<br>\r\nThus, 28 is a perfect number.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">2. Which of the following is the smallest perfect number?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">5<\/div><div class=\"spq_answer_block\" data-value=\"1\">6<\/div><div class=\"spq_answer_block\" data-value=\"2\">7<\/div><div class=\"spq_answer_block\" data-value=\"3\">8<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 6<br\/>The smallest perfect number is 6.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Which of the following is not a perfect number?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">27<\/div><div class=\"spq_answer_block\" data-value=\"1\">6<\/div><div class=\"spq_answer_block\" data-value=\"2\">28<\/div><div class=\"spq_answer_block\" data-value=\"3\">496<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 27<br\/>Factors of 27 are 1, 3, 9, and 27.<br>\r\n$1 + 3 + 9 = 13 \\neq 27$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">A perfect number can be written as the ____ of its proper factors.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">difference<\/div><div class=\"spq_answer_block\" data-value=\"1\">sum<\/div><div class=\"spq_answer_block\" data-value=\"2\">product<\/div><div class=\"spq_answer_block\" data-value=\"3\">none of these<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: sum<br\/>A perfect number can be written as the sum of its proper factors.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"What Are Perfect Numbers? 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Which of the following is the smallest perfect number?\",\n                    \"text\": \"2. Which of the following is the smallest perfect number?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The smallest perfect number is 6.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"5\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The smallest perfect number is 6.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"7\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The smallest perfect number is 6.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"8\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The smallest perfect number is 6.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"6\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The smallest perfect number is 6.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The smallest perfect number is 6.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is not a perfect number?\",\n                    \"text\": \"Which of the following is not a perfect number?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Factors of 27 are 1, 3, 9, and 27.<br>\r\n$$1 + 3 + 9 = 13 \\\\neq 27$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"6\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Factors of 27 are 1, 3, 9, and 27.<br>\r\n$$1 + 3 + 9 = 13 \\\\neq 27$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"28\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Factors of 27 are 1, 3, 9, and 27.<br>\r\n$$1 + 3 + 9 = 13 \\\\neq 27$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"496\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Factors of 27 are 1, 3, 9, and 27.<br>\r\n$$1 + 3 + 9 = 13 \\\\neq 27$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"27\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Factors of 27 are 1, 3, 9, and 27.<br>\r\n$$1 + 3 + 9 = 13 \\\\neq 27$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Factors of 27 are 1, 3, 9, and 27.<br>\r\n$$1 + 3 + 9 = 13 \\\\neq 27$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"A perfect number can be written as the ____ of its proper factors.\",\n                    \"text\": \"A perfect number can be written as the ____ of its proper factors.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"A perfect number can be written as the sum of its proper factors.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"difference\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A perfect number can be written as the sum of its proper factors.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"product\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A perfect number can be written as the sum of its proper factors.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"none of these\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A perfect number can be written as the sum of its proper factors.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"sum\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"A perfect number can be written as the sum of its proper factors.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"A perfect number can be written as the sum of its proper factors.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"10-frequently-asked-questions-on-perfect-numbers\">Frequently Asked Questions on Perfect Numbers<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-a86807fd-7747-4924-850d-ba2c11d58cea\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-a86807fd-7747-4924-850d-ba2c11d58cea\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a86807fd-7747-4924-850d-ba2c11d58cea\"><strong>What is the difference between perfect numbers and perfect squares?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-a86807fd-7747-4924-850d-ba2c11d58cea\">\n\n<p class=\"eplus-wrapper\">A perfect square is a number, which can be expressed as the square of a number from the same number system whereas a perfect number is a number, which can be expressed as the sum of its factors except itself.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-a86807fd-7747-4924-850d-ba2c11d58cea\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a86807fd-7747-4924-850d-ba2c11d58cea\"><strong>Are all perfect numbers even?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-a86807fd-7747-4924-850d-ba2c11d58cea\">\n\n<p class=\"eplus-wrapper\">The perfect numbers found till date have been even. There is no evidence of odd perfect numbers.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-a86807fd-7747-4924-850d-ba2c11d58cea\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a86807fd-7747-4924-850d-ba2c11d58cea\"><strong>How many perfect numbers are known?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-a86807fd-7747-4924-850d-ba2c11d58cea\">\n\n<p class=\"eplus-wrapper\">So far, 51 perfect numbers are known till date.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-a86807fd-7747-4924-850d-ba2c11d58cea\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a86807fd-7747-4924-850d-ba2c11d58cea\"><strong>Why is 1 not a perfect number?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-a86807fd-7747-4924-850d-ba2c11d58cea\">\n\n<p class=\"eplus-wrapper\">A perfect number can be written as the sum of its proper factors. The only factor of 1 is 1. So, 1 is not a perfect number.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-a86807fd-7747-4924-850d-ba2c11d58cea\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a86807fd-7747-4924-850d-ba2c11d58cea\"><strong>What are the first 5 perfect numbers?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-a86807fd-7747-4924-850d-ba2c11d58cea\">\n\n<p class=\"eplus-wrapper\">The first 5 perfect numbers are 6, 28, 496, 8128, and 33550336.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is a Perfect Number? Perfect numbers in math are the numbers (positive integers) that can be expressed as the sum of their factors (excluding the number itself). Factors of a number are the numbers that divide a given number exactly without leaving a remainder. A factor of a number is always less than or &#8230; <a title=\"What Are Perfect Numbers? Definition, List, Formula, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/perfect-numbers\" aria-label=\"More on What Are Perfect Numbers? Definition, List, Formula, Examples\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-27535","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/27535","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=27535"}],"version-history":[{"count":26,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/27535\/revisions"}],"predecessor-version":[{"id":35676,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/27535\/revisions\/35676"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=27535"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=27535"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=27535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}