\nLet the two positive integers be $x$ and $y$.
\nGiven LCM$(x, y) = 18$ and HCF$(x, y) = 3$.
\nUsing the property $x \\times y = \\text{LCM}(x,y) \\times \\text{HCF}(x,y)$:
\n$x \\times y = 18 \\times 3 = 54$.<\/p>\nSince the HCF is 3, both $x$ and $y$ must be multiples of 3. We can write $x = 3a$ and $y = 3b$, where $a$ and $b$ are positive integers.
\nSubstituting these into the product equation:
\n$(3a)(3b) = 54$
\n$9ab = 54$
\n$ab = 6$.<\/p>\n
Furthermore, the HCF of $x$ and $y$ is 3, which means HCF$(3a, 3b) = 3 \\times \\text{HCF}(a, b) = 3$. This implies HCF$(a, b) = 1$, i.e., $a$ and $b$ must be coprime.<\/p>\n
We need to find pairs of positive integers $(a, b)$ such that $ab=6$ and HCF$(a, b)=1$.
\nPossible pairs $(a,b)$ for $ab=6$:
\n1. (1, 6): HCF(1, 6) = 1. This pair is valid.
\n If $a=1, b=6$, then $x = 3 \\times 1 = 3$ and $y = 3 \\times 6 = 18$.
\n Check: HCF(3, 18) = 3, LCM(3, 18) = 18. Correct.
\n Sum $x+y = 3 + 18 = 21$.
\n2. (6, 1): HCF(6, 1) = 1. This pair is valid.
\n If $a=6, b=1$, then $x = 3 \\times 6 = 18$ and $y = 3 \\times 1 = 3$.
\n Check: HCF(18, 3) = 3, LCM(18, 3) = 18. Correct.
\n Sum $x+y = 18 + 3 = 21$.
\n3. (2, 3): HCF(2, 3) = 1. This pair is valid.
\n If $a=2, b=3$, then $x = 3 \\times 2 = 6$ and $y = 3 \\times 3 = 9$.
\n Check: HCF(6, 9) = 3, LCM(6, 9) = 18. Correct.
\n Sum $x+y = 6 + 9 = 15$.
\n4. (3, 2): HCF(3, 2) = 1. This pair is valid.
\n If $a=3, b=2$, then $x = 3 \\times 3 = 9$ and $y = 3 \\times 2 = 6$.
\n Check: HCF(9, 6) = 3, LCM(9, 6) = 18. Correct.
\n Sum $x+y = 9 + 6 = 15$.<\/p>\n
The possible sums of the two integers are 21 and 15.
\nThe minimum possible value of their sum is 15.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"
If the LCM and HCF of two positive integers are 18 and 3 respectively, then what is the minimum possible value of their sum ? [amp_mcq option1=”21″ option2=”15″ option3=”18″ option4=”16″ correct=”option2″] This question was previously asked in UPSC CISF-AC-EXE – 2021 Download PDFAttempt Online The minimum possible value of their sum is 15. For any … <\/p>\n
Detailed SolutionIf the LCM and HCF of two positive integers are 18 and 3 respectively,<\/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":[1089],"tags":[1110,1102],"class_list":["post-92930","post","type-post","status-publish","format-standard","hentry","category-upsc-cisf-ac-exe","tag-1110","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nIf the LCM and HCF of two positive integers are 18 and 3 respectively,<\/title>\n<meta name=\"description\" content=\"The minimum possible value of their sum is 15. For any two positive integers $x$ and $y$, the product of the integers is equal to the product of their Least Common Multiple (LCM) and Highest Common Factor (HCF): $x times y = text{LCM}(x,y) times text{HCF}(x,y)$. Also, both numbers must be multiples of their HCF.\" \/>\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\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If the LCM and HCF of two positive integers are 18 and 3 respectively,\" \/>\n<meta property=\"og:description\" content=\"The minimum possible value of their sum is 15. For any two positive integers $x$ and $y$, the product of the integers is equal to the product of their Least Common Multiple (LCM) and Highest Common Factor (HCF): $x times y = text{LCM}(x,y) times text{HCF}(x,y)$. Also, both numbers must be multiples of their HCF.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:37:00+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=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"If the LCM and HCF of two positive integers are 18 and 3 respectively,","description":"The minimum possible value of their sum is 15. For any two positive integers $x$ and $y$, the product of the integers is equal to the product of their Least Common Multiple (LCM) and Highest Common Factor (HCF): $x times y = text{LCM}(x,y) times text{HCF}(x,y)$. Also, both numbers must be multiples of their HCF.","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\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/","og_locale":"en_US","og_type":"article","og_title":"If the LCM and HCF of two positive integers are 18 and 3 respectively,","og_description":"The minimum possible value of their sum is 15. For any two positive integers $x$ and $y$, the product of the integers is equal to the product of their Least Common Multiple (LCM) and Highest Common Factor (HCF): $x times y = text{LCM}(x,y) times text{HCF}(x,y)$. Also, both numbers must be multiples of their HCF.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:37:00+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/","url":"https:\/\/exam.pscnotes.com\/mcq\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/","name":"If the LCM and HCF of two positive integers are 18 and 3 respectively,","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:37:00+00:00","dateModified":"2025-06-01T11:37:00+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The minimum possible value of their sum is 15. For any two positive integers $x$ and $y$, the product of the integers is equal to the product of their Least Common Multiple (LCM) and Highest Common Factor (HCF): $x \\times y = \\text{LCM}(x,y) \\times \\text{HCF}(x,y)$. Also, both numbers must be multiples of their HCF.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-lcm-and-hcf-of-two-positive-integers-are-18-and-3-respectively\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CISF-AC-EXE","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-cisf-ac-exe\/"},{"@type":"ListItem","position":3,"name":"If the LCM and HCF of two positive integers are 18 and 3 respectively,"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/92930","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=92930"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/92930\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=92930"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=92930"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=92930"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}