{"id":89940,"date":"2025-06-01T10:17:37","date_gmt":"2025-06-01T10:17:37","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=89940"},"modified":"2025-06-01T10:17:37","modified_gmt":"2025-06-01T10:17:37","slug":"how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/","title":{"rendered":"How many numbers from 1 to 1000 are there which are NOT divisible by a"},"content":{"rendered":"

How many numbers from 1 to 1000 are there which are NOT divisible by any of the digits 2, 3 and 5 ?<\/p>\n

[amp_mcq option1=”166″ option2=”266″ option3=”357″ option4=”366″ correct=”option2″]<\/p>\n

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This question was previously asked in<\/div>\n
UPSC CAPF – 2015<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
We need to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. This is equivalent to finding the total number of integers minus the number of integers that are divisible by at least one of 2, 3, or 5.
\nUsing the Principle of Inclusion-Exclusion:
\nTotal numbers = 1000
\nLet A be the set of numbers divisible by 2, B by 3, C by 5.
\n|A| = floor(1000\/2) = 500
\n|B| = floor(1000\/3) = 333
\n|C| = floor(1000\/5) = 200
\n|A \u2229 B| (divisible by 6) = floor(1000\/6) = 166
\n|A \u2229 C| (divisible by 10) = floor(1000\/10) = 100
\n|B \u2229 C| (divisible by 15) = floor(1000\/15) = 66
\n|A \u2229 B \u2229 C| (divisible by 30) = floor(1000\/30) = 33
\nNumber of elements divisible by at least one of 2, 3, or 5 is:
\n|A U B U C| = |A| + |B| + |C| – (|A \u2229 B| + |A \u2229 C| + |B \u2229 C|) + |A \u2229 B \u2229 C|
\n= 500 + 333 + 200 – (166 + 100 + 66) + 33
\n= 1033 – 332 + 33 = 701 + 33 = 734
\nThe number of numbers NOT divisible by any of the digits 2, 3, and 5 is:
\nTotal Numbers – |A U B U C| = 1000 – 734 = 266.
\n<\/section>\n
The problem requires finding numbers relatively prime to the product of the given digits (2*3*5=30) within a range. The Principle of Inclusion-Exclusion is a standard method for counting elements in the union of sets.
\n<\/section>\n
Alternatively, this can be viewed as finding numbers whose prime factors are not 2, 3, or 5. The fraction of such numbers up to N is approximately N * (1 – 1\/p\u2081) * (1 – 1\/p\u2082) * …, where p\u2081, p\u2082, … are the prime factors. For 30, the prime factors are 2, 3, 5. So the fraction is (1 – 1\/2)(1 – 1\/3)(1 – 1\/5) = (1\/2)(2\/3)(4\/5) = 8\/30 = 4\/15. Approximate count = 1000 * (4\/15) \u2248 266.67. The inclusion-exclusion method gives the exact integer count.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

How many numbers from 1 to 1000 are there which are NOT divisible by any of the digits 2, 3 and 5 ? [amp_mcq option1=”166″ option2=”266″ option3=”357″ option4=”366″ correct=”option2″] This question was previously asked in UPSC CAPF – 2015 Download PDFAttempt Online We need to find the number of integers from 1 to 1000 that … <\/p>\n

Detailed SolutionHow many numbers from 1 to 1000 are there which are NOT divisible by a<\/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":[1443,1102],"class_list":["post-89940","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1443","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nHow many numbers from 1 to 1000 are there which are NOT divisible by a<\/title>\n<meta name=\"description\" content=\"We need to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. This is equivalent to finding the total number of integers minus the number of integers that are divisible by at least one of 2, 3, or 5. Using the Principle of Inclusion-Exclusion: Total numbers = 1000 Let A be the set of numbers divisible by 2, B by 3, C by 5. |A| = floor(1000\/2) = 500 |B| = floor(1000\/3) = 333 |C| = floor(1000\/5) = 200 |A \u2229 B| (divisible by 6) = floor(1000\/6) = 166 |A \u2229 C| (divisible by 10) = floor(1000\/10) = 100 |B \u2229 C| (divisible by 15) = floor(1000\/15) = 66 |A \u2229 B \u2229 C| (divisible by 30) = floor(1000\/30) = 33 Number of elements divisible by at least one of 2, 3, or 5 is: |A U B U C| = |A| + |B| + |C| - (|A \u2229 B| + |A \u2229 C| + |B \u2229 C|) + |A \u2229 B \u2229 C| = 500 + 333 + 200 - (166 + 100 + 66) + 33 = 1033 - 332 + 33 = 701 + 33 = 734 The number of numbers NOT divisible by any of the digits 2, 3, and 5 is: Total Numbers - |A U B U C| = 1000 - 734 = 266. The problem requires finding numbers relatively prime to the product of the given digits (2*3*5=30) within a range. The Principle of Inclusion-Exclusion is a standard method for counting elements in the union of sets.\" \/>\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\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How many numbers from 1 to 1000 are there which are NOT divisible by a\" \/>\n<meta property=\"og:description\" content=\"We need to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. This is equivalent to finding the total number of integers minus the number of integers that are divisible by at least one of 2, 3, or 5. Using the Principle of Inclusion-Exclusion: Total numbers = 1000 Let A be the set of numbers divisible by 2, B by 3, C by 5. |A| = floor(1000\/2) = 500 |B| = floor(1000\/3) = 333 |C| = floor(1000\/5) = 200 |A \u2229 B| (divisible by 6) = floor(1000\/6) = 166 |A \u2229 C| (divisible by 10) = floor(1000\/10) = 100 |B \u2229 C| (divisible by 15) = floor(1000\/15) = 66 |A \u2229 B \u2229 C| (divisible by 30) = floor(1000\/30) = 33 Number of elements divisible by at least one of 2, 3, or 5 is: |A U B U C| = |A| + |B| + |C| - (|A \u2229 B| + |A \u2229 C| + |B \u2229 C|) + |A \u2229 B \u2229 C| = 500 + 333 + 200 - (166 + 100 + 66) + 33 = 1033 - 332 + 33 = 701 + 33 = 734 The number of numbers NOT divisible by any of the digits 2, 3, and 5 is: Total Numbers - |A U B U C| = 1000 - 734 = 266. The problem requires finding numbers relatively prime to the product of the given digits (2*3*5=30) within a range. The Principle of Inclusion-Exclusion is a standard method for counting elements in the union of sets.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:17:37+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":"How many numbers from 1 to 1000 are there which are NOT divisible by a","description":"We need to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. This is equivalent to finding the total number of integers minus the number of integers that are divisible by at least one of 2, 3, or 5. Using the Principle of Inclusion-Exclusion: Total numbers = 1000 Let A be the set of numbers divisible by 2, B by 3, C by 5. |A| = floor(1000\/2) = 500 |B| = floor(1000\/3) = 333 |C| = floor(1000\/5) = 200 |A \u2229 B| (divisible by 6) = floor(1000\/6) = 166 |A \u2229 C| (divisible by 10) = floor(1000\/10) = 100 |B \u2229 C| (divisible by 15) = floor(1000\/15) = 66 |A \u2229 B \u2229 C| (divisible by 30) = floor(1000\/30) = 33 Number of elements divisible by at least one of 2, 3, or 5 is: |A U B U C| = |A| + |B| + |C| - (|A \u2229 B| + |A \u2229 C| + |B \u2229 C|) + |A \u2229 B \u2229 C| = 500 + 333 + 200 - (166 + 100 + 66) + 33 = 1033 - 332 + 33 = 701 + 33 = 734 The number of numbers NOT divisible by any of the digits 2, 3, and 5 is: Total Numbers - |A U B U C| = 1000 - 734 = 266. The problem requires finding numbers relatively prime to the product of the given digits (2*3*5=30) within a range. The Principle of Inclusion-Exclusion is a standard method for counting elements in the union of sets.","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\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/","og_locale":"en_US","og_type":"article","og_title":"How many numbers from 1 to 1000 are there which are NOT divisible by a","og_description":"We need to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. This is equivalent to finding the total number of integers minus the number of integers that are divisible by at least one of 2, 3, or 5. Using the Principle of Inclusion-Exclusion: Total numbers = 1000 Let A be the set of numbers divisible by 2, B by 3, C by 5. |A| = floor(1000\/2) = 500 |B| = floor(1000\/3) = 333 |C| = floor(1000\/5) = 200 |A \u2229 B| (divisible by 6) = floor(1000\/6) = 166 |A \u2229 C| (divisible by 10) = floor(1000\/10) = 100 |B \u2229 C| (divisible by 15) = floor(1000\/15) = 66 |A \u2229 B \u2229 C| (divisible by 30) = floor(1000\/30) = 33 Number of elements divisible by at least one of 2, 3, or 5 is: |A U B U C| = |A| + |B| + |C| - (|A \u2229 B| + |A \u2229 C| + |B \u2229 C|) + |A \u2229 B \u2229 C| = 500 + 333 + 200 - (166 + 100 + 66) + 33 = 1033 - 332 + 33 = 701 + 33 = 734 The number of numbers NOT divisible by any of the digits 2, 3, and 5 is: Total Numbers - |A U B U C| = 1000 - 734 = 266. The problem requires finding numbers relatively prime to the product of the given digits (2*3*5=30) within a range. The Principle of Inclusion-Exclusion is a standard method for counting elements in the union of sets.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:17:37+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\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/","url":"https:\/\/exam.pscnotes.com\/mcq\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/","name":"How many numbers from 1 to 1000 are there which are NOT divisible by a","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:17:37+00:00","dateModified":"2025-06-01T10:17:37+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"We need to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. This is equivalent to finding the total number of integers minus the number of integers that are divisible by at least one of 2, 3, or 5. Using the Principle of Inclusion-Exclusion: Total numbers = 1000 Let A be the set of numbers divisible by 2, B by 3, C by 5. |A| = floor(1000\/2) = 500 |B| = floor(1000\/3) = 333 |C| = floor(1000\/5) = 200 |A \u2229 B| (divisible by 6) = floor(1000\/6) = 166 |A \u2229 C| (divisible by 10) = floor(1000\/10) = 100 |B \u2229 C| (divisible by 15) = floor(1000\/15) = 66 |A \u2229 B \u2229 C| (divisible by 30) = floor(1000\/30) = 33 Number of elements divisible by at least one of 2, 3, or 5 is: |A U B U C| = |A| + |B| + |C| - (|A \u2229 B| + |A \u2229 C| + |B \u2229 C|) + |A \u2229 B \u2229 C| = 500 + 333 + 200 - (166 + 100 + 66) + 33 = 1033 - 332 + 33 = 701 + 33 = 734 The number of numbers NOT divisible by any of the digits 2, 3, and 5 is: Total Numbers - |A U B U C| = 1000 - 734 = 266. The problem requires finding numbers relatively prime to the product of the given digits (2*3*5=30) within a range. The Principle of Inclusion-Exclusion is a standard method for counting elements in the union of sets.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/how-many-numbers-from-1-to-1000-are-there-which-are-not-divisible-by-a\/#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":"How many numbers from 1 to 1000 are there which are NOT divisible by a"}]},{"@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\/89940","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=89940"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/89940\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=89940"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=89940"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=89940"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}