{"id":10361,"date":"2022-04-02T21:19:25","date_gmt":"2022-04-02T15:49:25","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10361"},"modified":"2022-04-05T18:23:48","modified_gmt":"2022-04-05T12:53:48","slug":"use-euclids-division-lemma-to-show-that-the-square-of-any-positive-integer-is-either-of-the-form-3m-or-3m-1-for-some-integer-m","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/use-euclids-division-lemma-to-show-that-the-square-of-any-positive-integer-is-either-of-the-form-3m-or-3m-1-for-some-integer-m\/","title":{"rendered":"Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m."},"content":{"rendered":"

Solution :<\/h2>\n

By Euclid’s Division Algorithm, we have<\/p>\n

a = bq + r\u00a0 \u00a0 \u00a0 \u00a0…………..(i)<\/p>\n

On putting b = 3 in (1), we get<\/p>\n

a = 3q + r,\u00a0 \u00a0 \u00a0 [0 \\(\\le\\) r < 3]<\/p>\n

If r = 0\u00a0 \u00a0a = 3q\u00a0 \\(\\implies\\)\u00a0 \\(a^2\\) = \\(9q^2\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0…..(2)<\/p>\n

If r = 1\u00a0 a = 3q + 1\u00a0 \u00a0\\(\\implies\\)\u00a0 \\(a^2\\) = \\(9q^2 + 6q + 1\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0…..(3)<\/p>\n

If r = 2\u00a0 \u00a0a = 3q + 2\u00a0 \u00a0 \\(\\implies\\)\u00a0 \\(a^2\\) = \\(9q^2 + 12q + 4\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0…..(4)<\/p>\n

From (2), \\(9q^2\\) is a square of the form 3m, where m = \\(3q^2\\)<\/p>\n

From (3), \\(9q^2 + 6q + 1\\)\u00a0 i.e. , 3(\\(3q^2 + 2q\\)) + 1 is a square which is of the form 3m + 1 where m = \\(3q^2 +2q\\)<\/p>\n

From (4), \\(9q^2 + 12q + 14\\)\u00a0 i.e. , 3(\\(3q^2 + 2q + 1\\)) + 1 is a square which is of the form 3m + 1 where m = \\(3q^2 +4q + 1\\)<\/p>\n

Therefore, the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : By Euclid’s Division Algorithm, we have a = bq + r\u00a0 \u00a0 \u00a0 \u00a0…………..(i) On putting b = 3 in (1), we get a = 3q + r,\u00a0 \u00a0 \u00a0 [0 \\(\\le\\) r < 3] If r = 0\u00a0 \u00a0a = 3q\u00a0 \\(\\implies\\)\u00a0 \\(a^2\\) = \\(9q^2\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …<\/p>\n

Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.<\/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 square of any positive integer is either of the form 3m or 3m + 1 for some integer m. - 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-square-of-any-positive-integer-is-either-of-the-form-3m-or-3m-1-for-some-integer-m\/\" \/>\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 square of any positive integer is either of the form 3m or 3m + 1 for some integer m. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : By Euclid’s Division Algorithm, we have a = bq + r\u00a0 \u00a0 \u00a0 \u00a0…………..(i) On putting b = 3 in (1), we get a = 3q + r,\u00a0 \u00a0 \u00a0 [0 (le) r < 3] If r = 0\u00a0 \u00a0a = 3q\u00a0 (implies)\u00a0 (a^2) = (9q^2)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 … Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. 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