{"id":90362,"date":"2025-06-01T10:27:04","date_gmt":"2025-06-01T10:27:04","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90362"},"modified":"2025-06-01T10:27:04","modified_gmt":"2025-06-01T10:27:04","slug":"what-is-the-natural-number-n-for-which-3%e2%81%b9-3%c2%b9%c2%b2-3%e2%81%bf-is-a-perfect-cube","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-natural-number-n-for-which-3%e2%81%b9-3%c2%b9%c2%b2-3%e2%81%bf-is-a-perfect-cube\/","title":{"rendered":"What is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube"},"content":{"rendered":"

What is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube of an integer ?<\/p>\n

[amp_mcq option1=”10″ option2=”11″ option3=”13″ option4=”14″ correct=”option3″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2019<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nBased on typical test scenarios and provided answer keys for this specific question, n=13 is indicated as the correct answer, although standard mathematical analysis does not yield a perfect cube for this value.
\n<\/section>\n
\nFor the expression $3^9 + 3^{12} + 3^n$ to be a perfect cube, say $K^3$, we can factor out the lowest power of 3 present. If $n \\ge 9$, the lowest power is $3^9$. The expression becomes $3^9(1 + 3^{12-9} + 3^{n-9}) = 3^9(1 + 3^3 + 3^{n-9}) = 3^9(28 + 3^{n-9})$. Since $3^9 = (3^3)^3$ is already a perfect cube, the term $(28 + 3^{n-9})$ must also be a perfect cube of an integer, say $m^3$.
\n<\/section>\n
\nWe need to find a natural number $n$ such that $28 + 3^{n-9} = m^3$ for some integer $m$. Since $n$ is a natural number, $n \\ge 1$. For $3^{n-9}$ to be an integer, $n-9 \\ge 0$, so $n \\ge 9$. Let $k = n-9$, so $k \\ge 0$. We test the options for $n$:
\nA) $n=10 \\implies k=1$: $28 + 3^1 = 31$ (Not a perfect cube)
\nB) $n=11 \\implies k=2$: $28 + 3^2 = 37$ (Not a perfect cube)
\nC) $n=13 \\implies k=4$: $28 + 3^4 = 28 + 81 = 109$ (Not a perfect cube)
\nD) $n=14 \\implies k=5$: $28 + 3^5 = 28 + 243 = 271$ (Not a perfect cube)
\nAlso, checking $k=0$ ($n=9$) gives $28+3^0=29$ (not a cube). Standard mathematical methods confirm that for integer $k \\ge 0$, $28+3^k$ is not a perfect cube. This strongly suggests that the question as stated, or the provided options\/answer, might be flawed. However, given that this is a multiple-choice question from a competitive exam context and ‘C’ is indicated as the correct answer elsewhere, it implies there might be an intended but mathematically incorrect premise or a non-obvious property, which cannot be rigorously derived based on standard number theory.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

What is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube of an integer ? [amp_mcq option1=”10″ option2=”11″ option3=”13″ option4=”14″ correct=”option3″] This question was previously asked in UPSC CAPF – 2019 Download PDFAttempt Online Based on typical test scenarios and provided answer keys for this specific question, n=13 is … <\/p>\n

Detailed SolutionWhat is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1119,1102],"class_list":["post-90362","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1119","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nWhat is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube<\/title>\n<meta name=\"description\" content=\"Based on typical test scenarios and provided answer keys for this specific question, n=13 is indicated as the correct answer, although standard mathematical analysis does not yield a perfect cube for this value. For the expression $3^9 + 3^{12} + 3^n$ to be a perfect cube, say $K^3$, we can factor out the lowest power of 3 present. If $n ge 9$, the lowest power is $3^9$. The expression becomes $3^9(1 + 3^{12-9} + 3^{n-9}) = 3^9(1 + 3^3 + 3^{n-9}) = 3^9(28 + 3^{n-9})$. Since $3^9 = (3^3)^3$ is already a perfect cube, the term $(28 + 3^{n-9})$ must also be a perfect cube of an integer, say $m^3$.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-natural-number-n-for-which-3\u2079-3\u00b9\u00b2-3\u207f-is-a-perfect-cube\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube\" \/>\n<meta property=\"og:description\" content=\"Based on typical test scenarios and provided answer keys for this specific question, n=13 is indicated as the correct answer, although standard mathematical analysis does not yield a perfect cube for this value. For the expression $3^9 + 3^{12} + 3^n$ to be a perfect cube, say $K^3$, we can factor out the lowest power of 3 present. If $n ge 9$, the lowest power is $3^9$. The expression becomes $3^9(1 + 3^{12-9} + 3^{n-9}) = 3^9(1 + 3^3 + 3^{n-9}) = 3^9(28 + 3^{n-9})$. Since $3^9 = (3^3)^3$ is already a perfect cube, the term $(28 + 3^{n-9})$ must also be a perfect cube of an integer, say $m^3$.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-natural-number-n-for-which-3\u2079-3\u00b9\u00b2-3\u207f-is-a-perfect-cube\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:27:04+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"What is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube","description":"Based on typical test scenarios and provided answer keys for this specific question, n=13 is indicated as the correct answer, although standard mathematical analysis does not yield a perfect cube for this value. For the expression $3^9 + 3^{12} + 3^n$ to be a perfect cube, say $K^3$, we can factor out the lowest power of 3 present. If $n ge 9$, the lowest power is $3^9$. The expression becomes $3^9(1 + 3^{12-9} + 3^{n-9}) = 3^9(1 + 3^3 + 3^{n-9}) = 3^9(28 + 3^{n-9})$. Since $3^9 = (3^3)^3$ is already a perfect cube, the term $(28 + 3^{n-9})$ must also be a perfect cube of an integer, say $m^3$.","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:\/\/exam.pscnotes.com\/mcq\/what-is-the-natural-number-n-for-which-3\u2079-3\u00b9\u00b2-3\u207f-is-a-perfect-cube\/","og_locale":"en_US","og_type":"article","og_title":"What is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube","og_description":"Based on typical test scenarios and provided answer keys for this specific question, n=13 is indicated as the correct answer, although standard mathematical analysis does not yield a perfect cube for this value. For the expression $3^9 + 3^{12} + 3^n$ to be a perfect cube, say $K^3$, we can factor out the lowest power of 3 present. If $n ge 9$, the lowest power is $3^9$. The expression becomes $3^9(1 + 3^{12-9} + 3^{n-9}) = 3^9(1 + 3^3 + 3^{n-9}) = 3^9(28 + 3^{n-9})$. Since $3^9 = (3^3)^3$ is already a perfect cube, the term $(28 + 3^{n-9})$ must also be a perfect cube of an integer, say $m^3$.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-natural-number-n-for-which-3\u2079-3\u00b9\u00b2-3\u207f-is-a-perfect-cube\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:27:04+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-natural-number-n-for-which-3%e2%81%b9-3%c2%b9%c2%b2-3%e2%81%bf-is-a-perfect-cube\/","url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-natural-number-n-for-which-3%e2%81%b9-3%c2%b9%c2%b2-3%e2%81%bf-is-a-perfect-cube\/","name":"What is the natural number n for which 3\u2079 + 3\u00b9\u00b2 + 3\u207f is a perfect cube","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:27:04+00:00","dateModified":"2025-06-01T10:27:04+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"Based on typical test scenarios and provided answer keys for this specific question, n=13 is indicated as the correct answer, although standard mathematical analysis does not yield a perfect cube for this value. For the expression $3^9 + 3^{12} + 3^n$ to be a perfect cube, say $K^3$, we can factor out the lowest power of 3 present. If $n \\ge 9$, the lowest power is $3^9$. The expression becomes $3^9(1 + 3^{12-9} + 3^{n-9}) = 3^9(1 + 3^3 + 3^{n-9}) = 3^9(28 + 3^{n-9})$. 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