{"id":92380,"date":"2025-06-01T11:22:32","date_gmt":"2025-06-01T11:22:32","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=92380"},"modified":"2025-06-01T11:22:32","modified_gmt":"2025-06-01T11:22:32","slug":"the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/","title":{"rendered":"The sum of two positive integers is 52 and their LCM is 168. What is t"},"content":{"rendered":"<p>The sum of two positive integers is 52 and their LCM is 168. What is the ratio between the numbers?<\/p>\n<p>[amp_mcq option1=&#8221;2:3&#8243; option2=&#8221;5:4&#8243; option3=&#8221;7:6&#8243; option4=&#8221;7:8&#8243; correct=&#8221;option3&#8243;]<\/p>\n<div class=\"psc-box-pyq-exam-year-detail\">\n<div class=\"pyq-exam\">\n<div class=\"psc-heading\">This question was previously asked in<\/div>\n<div class=\"psc-title line-ellipsis\">UPSC CISF-AC-EXE &#8211; 2017<\/div>\n<\/div>\n<div class=\"pyq-exam-psc-buttons\"><a href=\"\/pyq\/pyq-upsc-cisf-ac-exe-2017.pdf\" target=\"_blank\" class=\"psc-pdf-button\" rel=\"noopener\">Download PDF<\/a><a href=\"\/pyq-upsc-cisf-ac-exe-2017\" target=\"_blank\" class=\"psc-attempt-button\" rel=\"noopener\">Attempt Online<\/a><\/div>\n<\/div>\n<section id=\"pyq-correct-answer\">\nThe correct answer is C.<br \/>\n<\/section>\n<section id=\"pyq-key-points\">\nLet the two positive integers be x and y.<br \/>\nGiven: x + y = 52 and LCM(x, y) = 168.<br \/>\nLet g be the Greatest Common Divisor (GCD) of x and y. So, x = ga and y = gb, where a and b are coprime positive integers (gcd(a, b) = 1).<br \/>\nUsing the given information:<br \/>\n1. x + y = ga + gb = g(a + b) = 52.<br \/>\n2. LCM(x, y) = g * a * b = 168.<br \/>\n<\/section>\n<section id=\"pyq-additional-information\">\nFrom g(a + b) = 52, we know that g must be a divisor of 52. The divisors of 52 are 1, 2, 4, 13, 26, 52.<br \/>\nFrom gab = 168, we know that g must be a divisor of 168.<br \/>\nSo, g must be a common divisor of 52 and 168.<br \/>\nDivisors of 52: 1, 2, 4, 13, 26, 52<br \/>\nDivisors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168<br \/>\nCommon divisors of 52 and 168 are 1, 2, 4. So g can be 1, 2, or 4.<br \/>\nWe also have the constraints: a and b are positive integers and gcd(a, b) = 1.<\/p>\n<p>Case 1: g = 1<br \/>\na + b = 52\/1 = 52<br \/>\nab = 168\/1 = 168<br \/>\nWe look for two coprime numbers a and b such that a+b=52 and ab=168. This is equivalent to solving the quadratic equation t\u00b2 &#8211; 52t + 168 = 0. The discriminant is 52\u00b2 &#8211; 4*168 = 2704 &#8211; 672 = 2032, which is not a perfect square, so integer solutions for t (a, b) do not exist. Alternatively, list factor pairs of 168 and check sums: (1, 168) sum 169; (2, 84) sum 86; (3, 56) sum 59; (4, 42) sum 46; (6, 28) sum 34; (7, 24) sum 31; (8, 21) sum 29; (12, 14) sum 26. None sum to 52.<\/p>\n<p>Case 2: g = 2<br \/>\na + b = 52\/2 = 26<br \/>\nab = 168\/2 = 84<br \/>\nWe look for two coprime numbers a and b such that a+b=26 and ab=84. Factors of 84: (1, 84) sum 85, gcd 1 (valid); (2, 42) sum 44, gcd 2 (invalid); (3, 28) sum 31, gcd 1 (valid); (4, 21) sum 25, gcd 1 (valid); (6, 14) sum 20, gcd 2 (invalid); (7, 12) sum 19, gcd 1 (valid). None sum to 26.<\/p>\n<p>Case 3: g = 4<br \/>\na + b = 52\/4 = 13<br \/>\nab = 168\/4 = 42<br \/>\nWe look for two coprime numbers a and b such that a+b=13 and ab=42. Factors of 42: (1, 42) sum 43, gcd 1 (valid); (2, 21) sum 23, gcd 1 (valid); (3, 14) sum 17, gcd 1 (valid); (6, 7) sum 13, gcd 1 (valid).<br \/>\nThe pair (6, 7) satisfies a+b=13 and gcd(6, 7)=1.<br \/>\nSo, possible values for (a, b) are (6, 7) or (7, 6).<br \/>\nIf (a, b) = (6, 7), then x = g*a = 4*6 = 24 and y = g*b = 4*7 = 28.<br \/>\nCheck: 24 + 28 = 52. LCM(24, 28) = LCM(2\u00b3*3, 2\u00b2*7) = 2\u00b3*3*7 = 8*21 = 168. This is correct.<br \/>\nThe two numbers are 24 and 28.<br \/>\nThe ratio between the numbers is 24:28 or 28:24.<br \/>\n24:28 simplifies to 6:7 (dividing by 4).<br \/>\n28:24 simplifies to 7:6 (dividing by 4).<br \/>\nOption C is 7:6.<\/p>\n<p>Therefore, the ratio between the numbers is 7:6 or 6:7. Option C provides 7:6.<br \/>\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>The sum of two positive integers is 52 and their LCM is 168. What is the ratio between the numbers? [amp_mcq option1=&#8221;2:3&#8243; option2=&#8221;5:4&#8243; option3=&#8221;7:6&#8243; option4=&#8221;7:8&#8243; correct=&#8221;option3&#8243;] This question was previously asked in UPSC CISF-AC-EXE &#8211; 2017 Download PDFAttempt Online The correct answer is C. Let the two positive integers be x and y. Given: x &#8230; <\/p>\n<p class=\"read-more-container\"><a title=\"The sum of two positive integers is 52 and their LCM is 168. What is t\" class=\"read-more button\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/#more-92380\">Detailed Solution<span class=\"screen-reader-text\">The sum of two positive integers is 52 and their LCM is 168. What is t<\/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":[1101,1102],"class_list":["post-92380","post","type-post","status-publish","format-standard","hentry","category-upsc-cisf-ac-exe","tag-1101","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The sum of two positive integers is 52 and their LCM is 168. What is t<\/title>\n<meta name=\"description\" content=\"The correct answer is C. Let the two positive integers be x and y. Given: x + y = 52 and LCM(x, y) = 168. Let g be the Greatest Common Divisor (GCD) of x and y. So, x = ga and y = gb, where a and b are coprime positive integers (gcd(a, b) = 1). Using the given information: 1. x + y = ga + gb = g(a + b) = 52. 2. LCM(x, y) = g * a * b = 168.\" \/>\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\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The sum of two positive integers is 52 and their LCM is 168. What is t\" \/>\n<meta property=\"og:description\" content=\"The correct answer is C. Let the two positive integers be x and y. Given: x + y = 52 and LCM(x, y) = 168. Let g be the Greatest Common Divisor (GCD) of x and y. So, x = ga and y = gb, where a and b are coprime positive integers (gcd(a, b) = 1). Using the given information: 1. x + y = ga + gb = g(a + b) = 52. 2. LCM(x, y) = g * a * b = 168.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:22:32+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":"The sum of two positive integers is 52 and their LCM is 168. What is t","description":"The correct answer is C. Let the two positive integers be x and y. Given: x + y = 52 and LCM(x, y) = 168. Let g be the Greatest Common Divisor (GCD) of x and y. So, x = ga and y = gb, where a and b are coprime positive integers (gcd(a, b) = 1). Using the given information: 1. x + y = ga + gb = g(a + b) = 52. 2. LCM(x, y) = g * a * b = 168.","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\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/","og_locale":"en_US","og_type":"article","og_title":"The sum of two positive integers is 52 and their LCM is 168. What is t","og_description":"The correct answer is C. Let the two positive integers be x and y. Given: x + y = 52 and LCM(x, y) = 168. Let g be the Greatest Common Divisor (GCD) of x and y. So, x = ga and y = gb, where a and b are coprime positive integers (gcd(a, b) = 1). Using the given information: 1. x + y = ga + gb = g(a + b) = 52. 2. LCM(x, y) = g * a * b = 168.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:22:32+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\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/","name":"The sum of two positive integers is 52 and their LCM is 168. What is t","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:22:32+00:00","dateModified":"2025-06-01T11:22:32+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is C. Let the two positive integers be x and y. Given: x + y = 52 and LCM(x, y) = 168. Let g be the Greatest Common Divisor (GCD) of x and y. So, x = ga and y = gb, where a and b are coprime positive integers (gcd(a, b) = 1). Using the given information: 1. x + y = ga + gb = g(a + b) = 52. 2. LCM(x, y) = g * a * b = 168.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-two-positive-integers-is-52-and-their-lcm-is-168-what-is-t\/#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":"The sum of two positive integers is 52 and their LCM is 168. 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