{"id":10343,"date":"2022-04-01T17:28:57","date_gmt":"2022-04-01T11:58:57","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10343"},"modified":"2022-04-05T18:24:05","modified_gmt":"2022-04-05T12:54:05","slug":"show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/","title":{"rendered":"Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>By Euclid&#8217;s division algorithm, we have<\/p>\n<p>a = bq + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(i)<\/p>\n<p>On putting, b = 6 in (1), we get<\/p>\n<p>a = 6q + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0 \\(\\le\\) r &lt; 6]<\/p>\n<p>If r = 0, a = 6q, 6q is divisible by 6\u00a0 \\(\\implies\\)\u00a0 6q is even.<\/p>\n<p>If r = 1, a = 6q + 1, 6q + 1 is not divisible by 2.<\/p>\n<p>If r = 2, a = 6q + 2, 6q + 2 is divisible by 2\u00a0 \\(\\implies\\)\u00a0 6q + 2 is even.<\/p>\n<p>If r = 3, a = 6q + 3, 6q + 3 is not divisible by 2.<\/p>\n<p>If r = 4, a = 6q + 4, 6q + 4 is divisible by 2\u00a0 \\(\\implies\\)\u00a0 6q + 4 is even.<\/p>\n<p>If r = 5, a = 6q + 5, 6q + 5 is not divisible by 2.<\/p>\n<p>Since, 6q, 6q + 2, 6q + 4 are even.<\/p>\n<p>Hence, <strong>the remaining integers are 6q + 1, 6q + 3, 6q + 5 are odd.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : By Euclid&#8217;s division algorithm, we have a = bq + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(i) On putting, b = 6 in (1), we get a = 6q + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0 \\(\\le\\) r &lt; 6] If r = 0, a = 6q, 6q is &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/\"> <span class=\"screen-reader-text\">Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer.<\/span> Read More &raquo;<\/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":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer. - 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\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : By Euclid&#8217;s division algorithm, we have a = bq + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(i) On putting, b = 6 in (1), we get a = 6q + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0 (le) r &lt; 6] If r = 0, a = 6q, 6q is &hellip; Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer. Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-04-01T11:58:57+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-04-05T12:54:05+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer.\",\"datePublished\":\"2022-04-01T11:58:57+00:00\",\"dateModified\":\"2022-04-05T12:54:05+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/\"},\"wordCount\":122,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Maths Questions\",\"Real Numbers Questions\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/\",\"url\":\"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/\",\"name\":\"Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer. - 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Mathemerize","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/show-that-any-positive-odd-integer-is-of-the-form-6q1-or-6q3-or-6q5-where-q-is-some-integer\/","og_locale":"en_US","og_type":"article","og_title":"Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer. - Mathemerize","og_description":"Solution : By Euclid&#8217;s division algorithm, we have a = bq + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8230;&#8230;&#8230;.(i) On putting, b = 6 in (1), we get a = 6q + r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0 (le) r &lt; 6] If r = 0, a = 6q, 6q is &hellip; Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer. 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