{"id":10363,"date":"2022-04-02T21:31:38","date_gmt":"2022-04-02T16:01:38","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10363"},"modified":"2022-04-05T18:23:39","modified_gmt":"2022-04-05T12:53:39","slug":"use-euclids-division-lemma-to-show-that-the-cube-of-any-positive-integer-is-either-of-the-form-9m-or-9m-1-or-9m-8","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/use-euclids-division-lemma-to-show-that-the-cube-of-any-positive-integer-is-either-of-the-form-9m-or-9m-1-or-9m-8\/","title":{"rendered":"Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m or 9m + 1 or 9m + 8."},"content":{"rendered":"

Solution :<\/h2>\n

Let m be any positive integer. Then it is of the form 3m, 3m + 1 or 3m + 2.<\/p>\n

Now, we have to prove that the cube of these can be rewritten in the form 9q, 9q + 1 or 9q + 8.<\/p>\n

Now,<\/p>\n

\\((3m)^3\\) = \\(27m^3\\) = \\(9(m^3)\\)<\/p>\n

= 9q, where q = \\(3m^3\\)<\/p>\n

\\((3m + 1)^3\\) = \\((3m)^3\\) + \\(3(3m)^2\\).1 + 3(3m).\\(1^2\\) + 1<\/p>\n

= \\(27m^3\\) + \\(27m^2\\) + 9m +1<\/p>\n

= \\(9(3m^3 + 3m^2 + m)\\) + 1<\/p>\n

= 9q + 1, where 1 = \\(3m^3 + 3m^2 + m\\)<\/p>\n

and \\((3m + 1)^3\\) = \\((3m)^3\\) + \\(3(3m)^2\\).2 + 3(3m).\\(2^2\\) + 8<\/p>\n

= \\(27m^3\\) + \\(54m^2\\) + 36m + 8<\/p>\n

= \\(9(3m^3 + 6m^2 + 4m)\\) + 8<\/p>\n

= 9q + 8, where q = \\(3m^3 + 6m^2 + 4m\\)<\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : Let m be any positive integer. Then it is of the form 3m, 3m + 1 or 3m + 2. Now, we have to prove that the cube of these can be rewritten in the form 9q, 9q + 1 or 9q + 8. Now, \\((3m)^3\\) = \\(27m^3\\) = \\(9(m^3)\\) = 9q, where …<\/p>\n

Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m or 9m + 1 or 9m + 8.<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,909],"tags":[],"yoast_head":"\nUse Euclid's Division Lemma to show that the cube of any positive integer is either of the form 9m or 9m + 1 or 9m + 8. - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/use-euclids-division-lemma-to-show-that-the-cube-of-any-positive-integer-is-either-of-the-form-9m-or-9m-1-or-9m-8\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Use Euclid's Division Lemma to show that the cube of any positive integer is either of the form 9m or 9m + 1 or 9m + 8. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : Let m be any positive integer. 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