{"id":90249,"date":"2025-06-01T10:23:53","date_gmt":"2025-06-01T10:23:53","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90249"},"modified":"2025-06-01T10:23:53","modified_gmt":"2025-06-01T10:23:53","slug":"suppose-x-y-z-are-three-positive-integers-such-that-x-%e2%89%a4-y-%e2%89%a4-z-and-xy","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/suppose-x-y-z-are-three-positive-integers-such-that-x-%e2%89%a4-y-%e2%89%a4-z-and-xy\/","title":{"rendered":"Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy"},"content":{"rendered":"

Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xyz = 72. Which one of the following values of S yields more than one solution to the equation x + y + z = S?<\/p>\n

[amp_mcq option1=”13″ option2=”14″ option3=”15″ option4=”16″ correct=”option2″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2018<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe value of S that yields more than one solution to the equation x + y + z = S, under the given conditions, is 14.
\n<\/section>\n
\n– We are looking for positive integer solutions (x, y, z) to xyz = 72 such that x \u2264 y \u2264 z.
\n– We need to find which sum S = x+y+z corresponds to more than one such distinct tuple (x, y, z).
\n– Let’s list all valid tuples (x, y, z) and their corresponding sums S:
\n 1. (1, 1, 72): S = 1+1+72 = 74
\n 2. (1, 2, 36): S = 1+2+36 = 39
\n 3. (1, 3, 24): S = 1+3+24 = 28
\n 4. (1, 4, 18): S = 1+4+18 = 23
\n 5. (1, 6, 12): S = 1+6+12 = 19
\n 6. (1, 8, 9): S = 1+8+9 = 18
\n 7. (2, 2, 18): S = 2+2+18 = 22 (since 2*18=36 and 2*2*18=72, 2<=2<=18)\n 8. (2, 3, 12): S = 2+3+12 = 17 (since 3*12=36 and 2*3*12=72, 2<=3<=12)\n 9. (2, 4, 9): S = 2+4+9 = 15 (since 4*9=36 and 2*4*9=72, 2<=4<=9)\n 10. (2, 6, 6): S = 2+6+6 = 14 (since 6*6=36 and 2*6*6=72, 2<=6<=6)\n 11. (3, 3, 8): S = 3+3+8 = 14 (since 3*8=24 and 3*3*8=72, 3<=3<=8)\n 12. (3, 4, 6): S = 3+4+6 = 13 (since 4*6=24 and 3*4*6=72, 3<=4<=6)\n- Checking for x=4: xyz=72 => yz=18, 4 <= y <= z. Possible (y,z) pairs for yz=18, y<=z are (1,18), (2,9), (3,6). None satisfy y >= 4. No solution starts with x=4.
\n– Checking for x>=5: Smallest product with x>=5 is 5*5*z. 5*5*z=72 => 25z=72, no integer z. Or 5*y*z=72 with 5<=y<=z. Minimum product 5*5*z means z>=5. 5*5*5=125 > 72. So no solutions for x>=5.
\n– Now, examine the sums S from our list:
\n – S = 13 corresponds to (3, 4, 6) – 1 solution.
\n – S = 14 corresponds to (2, 6, 6) and (3, 3, 8) – 2 solutions.
\n – S = 15 corresponds to (2, 4, 9) – 1 solution.
\n – S = 16 does not appear in the list of sums for any valid (x,y,z) tuple.
\n– The value of S that yields more than one solution is 14.
\n<\/section>\n
\nThis problem requires systematically finding all factorizations of 72 into three integers, ordering them, and then calculating their sums to identify duplicate sums. Ensuring the condition x \u2264 y \u2264 z is crucial to avoid counting permutations as distinct solutions.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xyz = 72. Which one of the following values of S yields more than one solution to the equation x + y + z = S? [amp_mcq option1=”13″ option2=”14″ option3=”15″ option4=”16″ correct=”option2″] This question was previously asked in UPSC … <\/p>\n

Detailed SolutionSuppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy<\/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":[1114,1102],"class_list":["post-90249","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1114","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nSuppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy<\/title>\n<meta name=\"description\" content=\"The value of S that yields more than one solution to the equation x + y + z = S, under the given conditions, is 14. - We are looking for positive integer solutions (x, y, z) to xyz = 72 such that x \u2264 y \u2264 z. - We need to find which sum S = x+y+z corresponds to more than one such distinct tuple (x, y, z). - Let's list all valid tuples (x, y, z) and their corresponding sums S: 1. (1, 1, 72): S = 1+1+72 = 74 2. (1, 2, 36): S = 1+2+36 = 39 3. (1, 3, 24): S = 1+3+24 = 28 4. (1, 4, 18): S = 1+4+18 = 23 5. (1, 6, 12): S = 1+6+12 = 19 6. (1, 8, 9): S = 1+8+9 = 18 7. (2, 2, 18): S = 2+2+18 = 22 (since 2*18=36 and 2*2*18=72, 2<=2<=18) 8. (2, 3, 12): S = 2+3+12 = 17 (since 3*12=36 and 2*3*12=72, 2<=3<=12) 9. (2, 4, 9): S = 2+4+9 = 15 (since 4*9=36 and 2*4*9=72, 2<=4<=9) 10. (2, 6, 6): S = 2+6+6 = 14 (since 6*6=36 and 2*6*6=72, 2<=6<=6) 11. (3, 3, 8): S = 3+3+8 = 14 (since 3*8=24 and 3*3*8=72, 3<=3<=8) 12. (3, 4, 6): S = 3+4+6 = 13 (since 4*6=24 and 3*4*6=72, 3<=4 yz=18, 4 <= y <= z. Possible (y,z) pairs for yz=18, y= 4. No solution starts with x=4. - Checking for x>=5: Smallest product with x>=5 is 5*5*z. 5*5*z=72 => 25z=72, no integer z. Or 5*y*z=72 with 5<=y=5. 5*5*5=125 > 72. So no solutions for x>=5. - Now, examine the sums S from our list: - S = 13 corresponds to (3, 4, 6) - 1 solution. - S = 14 corresponds to (2, 6, 6) and (3, 3, 8) - 2 solutions. - S = 15 corresponds to (2, 4, 9) - 1 solution. - S = 16 does not appear in the list of sums for any valid (x,y,z) tuple. - The value of S that yields more than one solution is 14.\" \/>\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\/suppose-x-y-z-are-three-positive-integers-such-that-x-\u2264-y-\u2264-z-and-xy\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy\" \/>\n<meta property=\"og:description\" content=\"The value of S that yields more than one solution to the equation x + y + z = S, under the given conditions, is 14. - We are looking for positive integer solutions (x, y, z) to xyz = 72 such that x \u2264 y \u2264 z. - We need to find which sum S = x+y+z corresponds to more than one such distinct tuple (x, y, z). - Let's list all valid tuples (x, y, z) and their corresponding sums S: 1. (1, 1, 72): S = 1+1+72 = 74 2. (1, 2, 36): S = 1+2+36 = 39 3. (1, 3, 24): S = 1+3+24 = 28 4. (1, 4, 18): S = 1+4+18 = 23 5. (1, 6, 12): S = 1+6+12 = 19 6. (1, 8, 9): S = 1+8+9 = 18 7. (2, 2, 18): S = 2+2+18 = 22 (since 2*18=36 and 2*2*18=72, 2<=2<=18) 8. (2, 3, 12): S = 2+3+12 = 17 (since 3*12=36 and 2*3*12=72, 2<=3<=12) 9. (2, 4, 9): S = 2+4+9 = 15 (since 4*9=36 and 2*4*9=72, 2<=4<=9) 10. (2, 6, 6): S = 2+6+6 = 14 (since 6*6=36 and 2*6*6=72, 2<=6<=6) 11. (3, 3, 8): S = 3+3+8 = 14 (since 3*8=24 and 3*3*8=72, 3<=3<=8) 12. (3, 4, 6): S = 3+4+6 = 13 (since 4*6=24 and 3*4*6=72, 3<=4 yz=18, 4 <= y <= z. Possible (y,z) pairs for yz=18, y= 4. No solution starts with x=4. - Checking for x>=5: Smallest product with x>=5 is 5*5*z. 5*5*z=72 => 25z=72, no integer z. Or 5*y*z=72 with 5<=y=5. 5*5*5=125 > 72. So no solutions for x>=5. - Now, examine the sums S from our list: - S = 13 corresponds to (3, 4, 6) - 1 solution. - S = 14 corresponds to (2, 6, 6) and (3, 3, 8) - 2 solutions. - S = 15 corresponds to (2, 4, 9) - 1 solution. - S = 16 does not appear in the list of sums for any valid (x,y,z) tuple. - The value of S that yields more than one solution is 14.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/suppose-x-y-z-are-three-positive-integers-such-that-x-\u2264-y-\u2264-z-and-xy\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:23:53+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":"Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy","description":"The value of S that yields more than one solution to the equation x + y + z = S, under the given conditions, is 14. - We are looking for positive integer solutions (x, y, z) to xyz = 72 such that x \u2264 y \u2264 z. - We need to find which sum S = x+y+z corresponds to more than one such distinct tuple (x, y, z). - Let's list all valid tuples (x, y, z) and their corresponding sums S: 1. (1, 1, 72): S = 1+1+72 = 74 2. (1, 2, 36): S = 1+2+36 = 39 3. (1, 3, 24): S = 1+3+24 = 28 4. (1, 4, 18): S = 1+4+18 = 23 5. (1, 6, 12): S = 1+6+12 = 19 6. (1, 8, 9): S = 1+8+9 = 18 7. (2, 2, 18): S = 2+2+18 = 22 (since 2*18=36 and 2*2*18=72, 2<=2<=18) 8. (2, 3, 12): S = 2+3+12 = 17 (since 3*12=36 and 2*3*12=72, 2<=3<=12) 9. (2, 4, 9): S = 2+4+9 = 15 (since 4*9=36 and 2*4*9=72, 2<=4<=9) 10. (2, 6, 6): S = 2+6+6 = 14 (since 6*6=36 and 2*6*6=72, 2<=6<=6) 11. (3, 3, 8): S = 3+3+8 = 14 (since 3*8=24 and 3*3*8=72, 3<=3<=8) 12. (3, 4, 6): S = 3+4+6 = 13 (since 4*6=24 and 3*4*6=72, 3<=4 yz=18, 4 <= y <= z. Possible (y,z) pairs for yz=18, y= 4. No solution starts with x=4. - Checking for x>=5: Smallest product with x>=5 is 5*5*z. 5*5*z=72 => 25z=72, no integer z. Or 5*y*z=72 with 5<=y=5. 5*5*5=125 > 72. So no solutions for x>=5. - Now, examine the sums S from our list: - S = 13 corresponds to (3, 4, 6) - 1 solution. - S = 14 corresponds to (2, 6, 6) and (3, 3, 8) - 2 solutions. - S = 15 corresponds to (2, 4, 9) - 1 solution. - S = 16 does not appear in the list of sums for any valid (x,y,z) tuple. - The value of S that yields more than one solution is 14.","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\/suppose-x-y-z-are-three-positive-integers-such-that-x-\u2264-y-\u2264-z-and-xy\/","og_locale":"en_US","og_type":"article","og_title":"Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy","og_description":"The value of S that yields more than one solution to the equation x + y + z = S, under the given conditions, is 14. - We are looking for positive integer solutions (x, y, z) to xyz = 72 such that x \u2264 y \u2264 z. - We need to find which sum S = x+y+z corresponds to more than one such distinct tuple (x, y, z). - Let's list all valid tuples (x, y, z) and their corresponding sums S: 1. (1, 1, 72): S = 1+1+72 = 74 2. (1, 2, 36): S = 1+2+36 = 39 3. (1, 3, 24): S = 1+3+24 = 28 4. (1, 4, 18): S = 1+4+18 = 23 5. (1, 6, 12): S = 1+6+12 = 19 6. (1, 8, 9): S = 1+8+9 = 18 7. (2, 2, 18): S = 2+2+18 = 22 (since 2*18=36 and 2*2*18=72, 2<=2<=18) 8. (2, 3, 12): S = 2+3+12 = 17 (since 3*12=36 and 2*3*12=72, 2<=3<=12) 9. (2, 4, 9): S = 2+4+9 = 15 (since 4*9=36 and 2*4*9=72, 2<=4<=9) 10. (2, 6, 6): S = 2+6+6 = 14 (since 6*6=36 and 2*6*6=72, 2<=6<=6) 11. (3, 3, 8): S = 3+3+8 = 14 (since 3*8=24 and 3*3*8=72, 3<=3<=8) 12. (3, 4, 6): S = 3+4+6 = 13 (since 4*6=24 and 3*4*6=72, 3<=4 yz=18, 4 <= y <= z. Possible (y,z) pairs for yz=18, y= 4. No solution starts with x=4. - Checking for x>=5: Smallest product with x>=5 is 5*5*z. 5*5*z=72 => 25z=72, no integer z. Or 5*y*z=72 with 5<=y=5. 5*5*5=125 > 72. So no solutions for x>=5. - Now, examine the sums S from our list: - S = 13 corresponds to (3, 4, 6) - 1 solution. - S = 14 corresponds to (2, 6, 6) and (3, 3, 8) - 2 solutions. - S = 15 corresponds to (2, 4, 9) - 1 solution. - S = 16 does not appear in the list of sums for any valid (x,y,z) tuple. - The value of S that yields more than one solution is 14.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/suppose-x-y-z-are-three-positive-integers-such-that-x-\u2264-y-\u2264-z-and-xy\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:23:53+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\/suppose-x-y-z-are-three-positive-integers-such-that-x-%e2%89%a4-y-%e2%89%a4-z-and-xy\/","url":"https:\/\/exam.pscnotes.com\/mcq\/suppose-x-y-z-are-three-positive-integers-such-that-x-%e2%89%a4-y-%e2%89%a4-z-and-xy\/","name":"Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:23:53+00:00","dateModified":"2025-06-01T10:23:53+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The value of S that yields more than one solution to the equation x + y + z = S, under the given conditions, is 14. - We are looking for positive integer solutions (x, y, z) to xyz = 72 such that x \u2264 y \u2264 z. - We need to find which sum S = x+y+z corresponds to more than one such distinct tuple (x, y, z). - Let's list all valid tuples (x, y, z) and their corresponding sums S: 1. (1, 1, 72): S = 1+1+72 = 74 2. (1, 2, 36): S = 1+2+36 = 39 3. (1, 3, 24): S = 1+3+24 = 28 4. (1, 4, 18): S = 1+4+18 = 23 5. (1, 6, 12): S = 1+6+12 = 19 6. (1, 8, 9): S = 1+8+9 = 18 7. (2, 2, 18): S = 2+2+18 = 22 (since 2*18=36 and 2*2*18=72, 2<=2<=18) 8. (2, 3, 12): S = 2+3+12 = 17 (since 3*12=36 and 2*3*12=72, 2<=3<=12) 9. (2, 4, 9): S = 2+4+9 = 15 (since 4*9=36 and 2*4*9=72, 2<=4<=9) 10. (2, 6, 6): S = 2+6+6 = 14 (since 6*6=36 and 2*6*6=72, 2<=6<=6) 11. (3, 3, 8): S = 3+3+8 = 14 (since 3*8=24 and 3*3*8=72, 3<=3<=8) 12. (3, 4, 6): S = 3+4+6 = 13 (since 4*6=24 and 3*4*6=72, 3<=4 yz=18, 4 <= y <= z. Possible (y,z) pairs for yz=18, y= 4. No solution starts with x=4. - Checking for x>=5: Smallest product with x>=5 is 5*5*z. 5*5*z=72 => 25z=72, no integer z. Or 5*y*z=72 with 5<=y=5. 5*5*5=125 > 72. So no solutions for x>=5. - Now, examine the sums S from our list: - S = 13 corresponds to (3, 4, 6) - 1 solution. - S = 14 corresponds to (2, 6, 6) and (3, 3, 8) - 2 solutions. - S = 15 corresponds to (2, 4, 9) - 1 solution. - S = 16 does not appear in the list of sums for any valid (x,y,z) tuple. - The value of S that yields more than one solution is 14.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/suppose-x-y-z-are-three-positive-integers-such-that-x-%e2%89%a4-y-%e2%89%a4-z-and-xy\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/suppose-x-y-z-are-three-positive-integers-such-that-x-%e2%89%a4-y-%e2%89%a4-z-and-xy\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/suppose-x-y-z-are-three-positive-integers-such-that-x-%e2%89%a4-y-%e2%89%a4-z-and-xy\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"Suppose x, y, z are three positive integers such that x \u2264 y \u2264 z and xy"}]},{"@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\/90249","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=90249"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90249\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=90249"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=90249"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=90249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}