{"id":92513,"date":"2025-06-01T11:27:07","date_gmt":"2025-06-01T11:27:07","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=92513"},"modified":"2025-06-01T11:27:07","modified_gmt":"2025-06-01T11:27:07","slug":"what-is-the-greatest-number-less-than-1000-which-when-divided-respecti","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/","title":{"rendered":"What is the greatest number less than 1000 which when divided respecti"},"content":{"rendered":"

What is the greatest number less than 1000 which when divided respectively by 5, 7 and 9 leaves the remainders 3, 5 and 7 respectively ?<\/p>\n

[amp_mcq option1=”943″ option2=”963″ option3=”953″ option4=”989″ 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 greatest number less than 1000 which satisfies the conditions is 943.
\n<\/section>\n
\nLet the number be N. The conditions are:
\nN \u2261 3 (mod 5)
\nN \u2261 5 (mod 7)
\nN \u2261 7 (mod 9)
\nNotice that in each case, the remainder is 2 less than the divisor. This means N + 2 is divisible by 5, 7, and 9.
\nThus, N + 2 must be a multiple of the Least Common Multiple (LCM) of 5, 7, and 9.
\nSince 5, 7, and 9 are pairwise coprime, LCM(5, 7, 9) = 5 * 7 * 9 = 315.
\nSo, N + 2 = 315k for some integer k.
\nN = 315k – 2.
\nWe are looking for the greatest number less than 1000.
\nFor k=1, N = 315(1) – 2 = 313.
\nFor k=2, N = 315(2) – 2 = 630 – 2 = 628.
\nFor k=3, N = 315(3) – 2 = 945 – 2 = 943.
\nFor k=4, N = 315(4) – 2 = 1260 – 2 = 1258, which is greater than 1000.
\nThe greatest number less than 1000 is 943.
\n<\/section>\n
\nThis problem is an application of the Chinese Remainder Theorem, but can be solved more directly by observing the pattern in the remainders. Checking the answer: 943 divided by 5 gives 188 with remainder 3. 943 divided by 7 gives 134 with remainder 5 (134 * 7 = 938). 943 divided by 9 gives 104 with remainder 7 (104 * 9 = 936). The conditions are satisfied.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

What is the greatest number less than 1000 which when divided respectively by 5, 7 and 9 leaves the remainders 3, 5 and 7 respectively ? [amp_mcq option1=”943″ option2=”963″ option3=”953″ option4=”989″ correct=”option1″] This question was previously asked in UPSC CISF-AC-EXE – 2018 Download PDFAttempt Online The greatest number less than 1000 which satisfies the conditions … <\/p>\n

Detailed SolutionWhat is the greatest number less than 1000 which when divided respecti<\/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-92513","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":"\nWhat is the greatest number less than 1000 which when divided respecti<\/title>\n<meta name=\"description\" content=\"The greatest number less than 1000 which satisfies the conditions is 943. Let the number be N. The conditions are: N \u2261 3 (mod 5) N \u2261 5 (mod 7) N \u2261 7 (mod 9) Notice that in each case, the remainder is 2 less than the divisor. This means N + 2 is divisible by 5, 7, and 9. Thus, N + 2 must be a multiple of the Least Common Multiple (LCM) of 5, 7, and 9. Since 5, 7, and 9 are pairwise coprime, LCM(5, 7, 9) = 5 * 7 * 9 = 315. So, N + 2 = 315k for some integer k. N = 315k - 2. We are looking for the greatest number less than 1000. For k=1, N = 315(1) - 2 = 313. For k=2, N = 315(2) - 2 = 630 - 2 = 628. For k=3, N = 315(3) - 2 = 945 - 2 = 943. For k=4, N = 315(4) - 2 = 1260 - 2 = 1258, which is greater than 1000. The greatest number less than 1000 is 943.\" \/>\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-greatest-number-less-than-1000-which-when-divided-respecti\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the greatest number less than 1000 which when divided respecti\" \/>\n<meta property=\"og:description\" content=\"The greatest number less than 1000 which satisfies the conditions is 943. Let the number be N. The conditions are: N \u2261 3 (mod 5) N \u2261 5 (mod 7) N \u2261 7 (mod 9) Notice that in each case, the remainder is 2 less than the divisor. This means N + 2 is divisible by 5, 7, and 9. Thus, N + 2 must be a multiple of the Least Common Multiple (LCM) of 5, 7, and 9. Since 5, 7, and 9 are pairwise coprime, LCM(5, 7, 9) = 5 * 7 * 9 = 315. So, N + 2 = 315k for some integer k. N = 315k - 2. We are looking for the greatest number less than 1000. For k=1, N = 315(1) - 2 = 313. For k=2, N = 315(2) - 2 = 630 - 2 = 628. For k=3, N = 315(3) - 2 = 945 - 2 = 943. For k=4, N = 315(4) - 2 = 1260 - 2 = 1258, which is greater than 1000. The greatest number less than 1000 is 943.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T11:27:07+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":"What is the greatest number less than 1000 which when divided respecti","description":"The greatest number less than 1000 which satisfies the conditions is 943. Let the number be N. The conditions are: N \u2261 3 (mod 5) N \u2261 5 (mod 7) N \u2261 7 (mod 9) Notice that in each case, the remainder is 2 less than the divisor. This means N + 2 is divisible by 5, 7, and 9. Thus, N + 2 must be a multiple of the Least Common Multiple (LCM) of 5, 7, and 9. Since 5, 7, and 9 are pairwise coprime, LCM(5, 7, 9) = 5 * 7 * 9 = 315. So, N + 2 = 315k for some integer k. N = 315k - 2. We are looking for the greatest number less than 1000. For k=1, N = 315(1) - 2 = 313. For k=2, N = 315(2) - 2 = 630 - 2 = 628. For k=3, N = 315(3) - 2 = 945 - 2 = 943. For k=4, N = 315(4) - 2 = 1260 - 2 = 1258, which is greater than 1000. The greatest number less than 1000 is 943.","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-greatest-number-less-than-1000-which-when-divided-respecti\/","og_locale":"en_US","og_type":"article","og_title":"What is the greatest number less than 1000 which when divided respecti","og_description":"The greatest number less than 1000 which satisfies the conditions is 943. Let the number be N. The conditions are: N \u2261 3 (mod 5) N \u2261 5 (mod 7) N \u2261 7 (mod 9) Notice that in each case, the remainder is 2 less than the divisor. This means N + 2 is divisible by 5, 7, and 9. Thus, N + 2 must be a multiple of the Least Common Multiple (LCM) of 5, 7, and 9. Since 5, 7, and 9 are pairwise coprime, LCM(5, 7, 9) = 5 * 7 * 9 = 315. So, N + 2 = 315k for some integer k. N = 315k - 2. We are looking for the greatest number less than 1000. For k=1, N = 315(1) - 2 = 313. For k=2, N = 315(2) - 2 = 630 - 2 = 628. For k=3, N = 315(3) - 2 = 945 - 2 = 943. For k=4, N = 315(4) - 2 = 1260 - 2 = 1258, which is greater than 1000. The greatest number less than 1000 is 943.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T11:27:07+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\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/","url":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/","name":"What is the greatest number less than 1000 which when divided respecti","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T11:27:07+00:00","dateModified":"2025-06-01T11:27:07+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The greatest number less than 1000 which satisfies the conditions is 943. Let the number be N. The conditions are: N \u2261 3 (mod 5) N \u2261 5 (mod 7) N \u2261 7 (mod 9) Notice that in each case, the remainder is 2 less than the divisor. This means N + 2 is divisible by 5, 7, and 9. Thus, N + 2 must be a multiple of the Least Common Multiple (LCM) of 5, 7, and 9. Since 5, 7, and 9 are pairwise coprime, LCM(5, 7, 9) = 5 * 7 * 9 = 315. So, N + 2 = 315k for some integer k. N = 315k - 2. We are looking for the greatest number less than 1000. For k=1, N = 315(1) - 2 = 313. For k=2, N = 315(2) - 2 = 630 - 2 = 628. For k=3, N = 315(3) - 2 = 945 - 2 = 943. For k=4, N = 315(4) - 2 = 1260 - 2 = 1258, which is greater than 1000. The greatest number less than 1000 is 943.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/what-is-the-greatest-number-less-than-1000-which-when-divided-respecti\/#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 greatest number less than 1000 which when divided respecti"}]},{"@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\/92513","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=92513"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/92513\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=92513"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=92513"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=92513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}