{"id":92518,"date":"2025-06-01T11:27:13","date_gmt":"2025-06-01T11:27:13","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=92518"},"modified":"2025-06-01T11:27:13","modified_gmt":"2025-06-01T11:27:13","slug":"the-difference-of-squares-of-two-consecutive-odd-numbers-is-always","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/","title":{"rendered":"The difference of squares of two consecutive odd numbers is always"},"content":{"rendered":"

The difference of squares of two consecutive odd numbers is always<\/p>\n

[amp_mcq option1=”divisible by 8″ option2=”divisible by 3″ option3=”divisible by 16″ option4=”None of the above” correct=”option1″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CISF-AC-EXE – 2018<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe difference of squares of two consecutive odd numbers is always divisible by 8.
\n<\/section>\n
\nLet the two consecutive odd numbers be represented by (2n + 1) and (2n + 3), where n is an integer.
\nThe difference of their squares is:
\n(2n + 3)^2 – (2n + 1)^2
\nUsing the algebraic identity a^2 – b^2 = (a – b)(a + b):
\n= [(2n + 3) – (2n + 1)] * [(2n + 3) + (2n + 1)]
\n= [2n + 3 – 2n – 1] * [2n + 3 + 2n + 1]
\n= [2] * [4n + 4]
\n= 2 * 4(n + 1)
\n= 8(n + 1)
\nSince n is an integer, (n + 1) is also an integer. Therefore, the expression 8(n + 1) is always a multiple of 8.
\n<\/section>\n
\nLet’s verify with examples:
\nConsecutive odd numbers 1 and 3: 3^2 – 1^2 = 9 – 1 = 8. 8 is divisible by 8.
\nConsecutive odd numbers 3 and 5: 5^2 – 3^2 = 25 – 9 = 16. 16 is divisible by 8.
\nConsecutive odd numbers 5 and 7: 7^2 – 5^2 = 49 – 25 = 24. 24 is divisible by 8.
\nThe difference is always in the form 8 * (an integer), so it is always divisible by 8.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

The difference of squares of two consecutive odd numbers is always [amp_mcq option1=”divisible by 8″ option2=”divisible by 3″ option3=”divisible by 16″ option4=”None of the above” correct=”option1″] This question was previously asked in UPSC CISF-AC-EXE – 2018 Download PDFAttempt Online The difference of squares of two consecutive odd numbers is always divisible by 8. Let the … <\/p>\n

Detailed SolutionThe difference of squares of two consecutive odd numbers is always<\/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":[1114,1102],"class_list":["post-92518","post","type-post","status-publish","format-standard","hentry","category-upsc-cisf-ac-exe","tag-1114","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nThe difference of squares of two consecutive odd numbers is always<\/title>\n<meta name=\"description\" content=\"The difference of squares of two consecutive odd numbers is always divisible by 8. Let the two consecutive odd numbers be represented by (2n + 1) and (2n + 3), where n is an integer. The difference of their squares is: (2n + 3)^2 - (2n + 1)^2 Using the algebraic identity a^2 - b^2 = (a - b)(a + b): = [(2n + 3) - (2n + 1)] * [(2n + 3) + (2n + 1)] = [2n + 3 - 2n - 1] * [2n + 3 + 2n + 1] = [2] * [4n + 4] = 2 * 4(n + 1) = 8(n + 1) Since n is an integer, (n + 1) is also an integer. Therefore, the expression 8(n + 1) is always a multiple of 8.\" \/>\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-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The difference of squares of two consecutive odd numbers is always\" \/>\n<meta property=\"og:description\" content=\"The difference of squares of two consecutive odd numbers is always divisible by 8. Let the two consecutive odd numbers be represented by (2n + 1) and (2n + 3), where n is an integer. The difference of their squares is: (2n + 3)^2 - (2n + 1)^2 Using the algebraic identity a^2 - b^2 = (a - b)(a + b): = [(2n + 3) - (2n + 1)] * [(2n + 3) + (2n + 1)] = [2n + 3 - 2n - 1] * [2n + 3 + 2n + 1] = [2] * [4n + 4] = 2 * 4(n + 1) = 8(n + 1) Since n is an integer, (n + 1) is also an integer. Therefore, the expression 8(n + 1) is always a multiple of 8.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:27:13+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":"The difference of squares of two consecutive odd numbers is always","description":"The difference of squares of two consecutive odd numbers is always divisible by 8. Let the two consecutive odd numbers be represented by (2n + 1) and (2n + 3), where n is an integer. The difference of their squares is: (2n + 3)^2 - (2n + 1)^2 Using the algebraic identity a^2 - b^2 = (a - b)(a + b): = [(2n + 3) - (2n + 1)] * [(2n + 3) + (2n + 1)] = [2n + 3 - 2n - 1] * [2n + 3 + 2n + 1] = [2] * [4n + 4] = 2 * 4(n + 1) = 8(n + 1) Since n is an integer, (n + 1) is also an integer. Therefore, the expression 8(n + 1) is always a multiple of 8.","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-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/","og_locale":"en_US","og_type":"article","og_title":"The difference of squares of two consecutive odd numbers is always","og_description":"The difference of squares of two consecutive odd numbers is always divisible by 8. Let the two consecutive odd numbers be represented by (2n + 1) and (2n + 3), where n is an integer. The difference of their squares is: (2n + 3)^2 - (2n + 1)^2 Using the algebraic identity a^2 - b^2 = (a - b)(a + b): = [(2n + 3) - (2n + 1)] * [(2n + 3) + (2n + 1)] = [2n + 3 - 2n - 1] * [2n + 3 + 2n + 1] = [2] * [4n + 4] = 2 * 4(n + 1) = 8(n + 1) Since n is an integer, (n + 1) is also an integer. Therefore, the expression 8(n + 1) is always a multiple of 8.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:27:13+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\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/","name":"The difference of squares of two consecutive odd numbers is always","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:27:13+00:00","dateModified":"2025-06-01T11:27:13+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The difference of squares of two consecutive odd numbers is always divisible by 8. Let the two consecutive odd numbers be represented by (2n + 1) and (2n + 3), where n is an integer. The difference of their squares is: (2n + 3)^2 - (2n + 1)^2 Using the algebraic identity a^2 - b^2 = (a - b)(a + b): = [(2n + 3) - (2n + 1)] * [(2n + 3) + (2n + 1)] = [2n + 3 - 2n - 1] * [2n + 3 + 2n + 1] = [2] * [4n + 4] = 2 * 4(n + 1) = 8(n + 1) Since n is an integer, (n + 1) is also an integer. Therefore, the expression 8(n + 1) is always a multiple of 8.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-difference-of-squares-of-two-consecutive-odd-numbers-is-always\/#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 difference of squares of two consecutive odd numbers is always"}]},{"@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\/92518","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=92518"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/92518\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=92518"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=92518"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=92518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}