{"id":92379,"date":"2025-06-01T11:22:32","date_gmt":"2025-06-01T11:22:32","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=92379"},"modified":"2025-06-01T11:22:32","modified_gmt":"2025-06-01T11:22:32","slug":"what-is-the-number-of-all-possible-positive-integer-values-of-n-for","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/","title":{"rendered":"What is the number of all possible positive integer values of ‘n’ for"},"content":{"rendered":"

What is the number of all possible positive integer values of ‘n’ for which n\u00b2 + 96 is a perfect square ?<\/p>\n

[amp_mcq option1=”2″ option2=”4″ option3=”5″ option4=”Infinite” correct=”option2″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CISF-AC-EXE – 2017<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is B.
\n<\/section>\n
\nWe are looking for positive integer values of ‘n’ such that n\u00b2 + 96 is a perfect square.
\nLet n\u00b2 + 96 = k\u00b2, where k is an integer.
\nSince n is a positive integer, n\u00b2 is positive, so k\u00b2 must be greater than n\u00b2. This implies k > n (assuming k is positive, which it must be since k\u00b2 = n\u00b2 + 96).
\nRearranging the equation, we get k\u00b2 – n\u00b2 = 96.
\nThis is a difference of squares, which can be factored as (k – n)(k + n) = 96.
\nSince k and n are integers, (k-n) and (k+n) must be integer factors of 96.
\nAlso, (k+n) – (k-n) = 2n. Since n is an integer, 2n is an even integer. This means (k-n) and (k+n) must have the same parity. Since their product (96) is even, both factors must be even.
\nFurthermore, since n is positive, k+n > k-n. Also, k+n > 0 (as n>0 and k>n).
\n<\/section>\n
\nWe need to find pairs of even factors (a, b) of 96 such that a * b = 96 and b > a.
\nThe factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.
\nThe even factors are 2, 4, 6, 8, 12, 16, 24, 32, 48, 96.
\nPairs (a, b) where a and b are even, a*b=96, and b > a:
\n1. (2, 48) -> k – n = 2, k + n = 48. Adding gives 2k=50, k=25. Subtracting gives 2n=46, n=23. (n=23 is a positive integer)
\n2. (4, 24) -> k – n = 4, k + n = 24. Adding gives 2k=28, k=14. Subtracting gives 2n=20, n=10. (n=10 is a positive integer)
\n3. (6, 16) -> k – n = 6, k + n = 16. Adding gives 2k=22, k=11. Subtracting gives 2n=10, n=5. (n=5 is a positive integer)
\n4. (8, 12) -> k – n = 8, k + n = 12. Adding gives 2k=20, k=10. Subtracting gives 2n=4, n=2. (n=2 is a positive integer)
\nWe found 4 distinct positive integer values for n: 23, 10, 5, and 2.
\nTherefore, there are 4 possible positive integer values of ‘n’ for which n\u00b2 + 96 is a perfect square.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

What is the number of all possible positive integer values of ‘n’ for which n\u00b2 + 96 is a perfect square ? [amp_mcq option1=”2″ option2=”4″ option3=”5″ option4=”Infinite” correct=”option2″] This question was previously asked in UPSC CISF-AC-EXE – 2017 Download PDFAttempt Online The correct answer is B. We are looking for positive integer values of ‘n’ … <\/p>\n

Detailed SolutionWhat is the number of all possible positive integer values of ‘n’ for<\/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-92379","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":"\nWhat is the number of all possible positive integer values of 'n' for<\/title>\n<meta name=\"description\" content=\"The correct answer is B. We are looking for positive integer values of 'n' such that n\u00b2 + 96 is a perfect square. Let n\u00b2 + 96 = k\u00b2, where k is an integer. Since n is a positive integer, n\u00b2 is positive, so k\u00b2 must be greater than n\u00b2. This implies k > n (assuming k is positive, which it must be since k\u00b2 = n\u00b2 + 96). Rearranging the equation, we get k\u00b2 - n\u00b2 = 96. This is a difference of squares, which can be factored as (k - n)(k + n) = 96. Since k and n are integers, (k-n) and (k+n) must be integer factors of 96. Also, (k+n) - (k-n) = 2n. Since n is an integer, 2n is an even integer. This means (k-n) and (k+n) must have the same parity. Since their product (96) is even, both factors must be even. Furthermore, since n is positive, k+n > k-n. Also, k+n > 0 (as n>0 and k>n).\" \/>\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-number-of-all-possible-positive-integer-values-of-n-for\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the number of all possible positive integer values of 'n' for\" \/>\n<meta property=\"og:description\" content=\"The correct answer is B. We are looking for positive integer values of 'n' such that n\u00b2 + 96 is a perfect square. Let n\u00b2 + 96 = k\u00b2, where k is an integer. Since n is a positive integer, n\u00b2 is positive, so k\u00b2 must be greater than n\u00b2. This implies k > n (assuming k is positive, which it must be since k\u00b2 = n\u00b2 + 96). Rearranging the equation, we get k\u00b2 - n\u00b2 = 96. This is a difference of squares, which can be factored as (k - n)(k + n) = 96. Since k and n are integers, (k-n) and (k+n) must be integer factors of 96. Also, (k+n) - (k-n) = 2n. Since n is an integer, 2n is an even integer. This means (k-n) and (k+n) must have the same parity. Since their product (96) is even, both factors must be even. Furthermore, since n is positive, k+n > k-n. Also, k+n > 0 (as n>0 and k>n).\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/\" \/>\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":"What is the number of all possible positive integer values of 'n' for","description":"The correct answer is B. We are looking for positive integer values of 'n' such that n\u00b2 + 96 is a perfect square. Let n\u00b2 + 96 = k\u00b2, where k is an integer. Since n is a positive integer, n\u00b2 is positive, so k\u00b2 must be greater than n\u00b2. This implies k > n (assuming k is positive, which it must be since k\u00b2 = n\u00b2 + 96). Rearranging the equation, we get k\u00b2 - n\u00b2 = 96. This is a difference of squares, which can be factored as (k - n)(k + n) = 96. Since k and n are integers, (k-n) and (k+n) must be integer factors of 96. Also, (k+n) - (k-n) = 2n. Since n is an integer, 2n is an even integer. This means (k-n) and (k+n) must have the same parity. Since their product (96) is even, both factors must be even. Furthermore, since n is positive, k+n > k-n. Also, k+n > 0 (as n>0 and k>n).","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-number-of-all-possible-positive-integer-values-of-n-for\/","og_locale":"en_US","og_type":"article","og_title":"What is the number of all possible positive integer values of 'n' for","og_description":"The correct answer is B. We are looking for positive integer values of 'n' such that n\u00b2 + 96 is a perfect square. Let n\u00b2 + 96 = k\u00b2, where k is an integer. Since n is a positive integer, n\u00b2 is positive, so k\u00b2 must be greater than n\u00b2. This implies k > n (assuming k is positive, which it must be since k\u00b2 = n\u00b2 + 96). Rearranging the equation, we get k\u00b2 - n\u00b2 = 96. This is a difference of squares, which can be factored as (k - n)(k + n) = 96. Since k and n are integers, (k-n) and (k+n) must be integer factors of 96. Also, (k+n) - (k-n) = 2n. Since n is an integer, 2n is an even integer. This means (k-n) and (k+n) must have the same parity. Since their product (96) is even, both factors must be even. Furthermore, since n is positive, k+n > k-n. Also, k+n > 0 (as n>0 and k>n).","og_url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/","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\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/","url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/","name":"What is the number of all possible positive integer values of 'n' for","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 B. We are looking for positive integer values of 'n' such that n\u00b2 + 96 is a perfect square. Let n\u00b2 + 96 = k\u00b2, where k is an integer. Since n is a positive integer, n\u00b2 is positive, so k\u00b2 must be greater than n\u00b2. This implies k > n (assuming k is positive, which it must be since k\u00b2 = n\u00b2 + 96). Rearranging the equation, we get k\u00b2 - n\u00b2 = 96. This is a difference of squares, which can be factored as (k - n)(k + n) = 96. Since k and n are integers, (k-n) and (k+n) must be integer factors of 96. Also, (k+n) - (k-n) = 2n. Since n is an integer, 2n is an even integer. This means (k-n) and (k+n) must have the same parity. Since their product (96) is even, both factors must be even. Furthermore, since n is positive, k+n > k-n. Also, k+n > 0 (as n>0 and k>n).","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-number-of-all-possible-positive-integer-values-of-n-for\/#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":"What is the number of all possible positive integer values of ‘n’ for"}]},{"@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\/92379","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=92379"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/92379\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=92379"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=92379"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=92379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}