{"id":57463,"date":"2025-08-22T13:40:17","date_gmt":"2025-08-22T08:10:17","guid":{"rendered":"https:\/\/e-gmat.com\/blogs\/?p=57463"},"modified":"2025-09-05T00:51:53","modified_gmt":"2025-09-04T19:21:53","slug":"how-many-positive-three-digit-integers-are-divisible-by-both","status":"publish","type":"post","link":"https:\/\/e-gmat.com\/blogs\/how-many-positive-three-digit-integers-are-divisible-by-both\/","title":{"rendered":"How many positive three-digit integers are divisible by both 3 and 4?"},"content":{"rendered":"A <\/span> 3<\/span> min read <\/span><\/span>\n

How many positive three-digit integers are divisible by both 3 and 4?<\/p>\n\n\n\n

A. 75<\/p>\n\n\n\n

B. 128<\/p>\n\n\n\n

C. 150<\/p>\n\n\n\n

D. 225<\/p>\n\n\n\n

E. 300<\/p>\n\n\n\n

Solution:<\/h2>\n\n\n\n
  1. Translate the problem requirements<\/strong>: We need to find three-digit numbers (100 to 999) that are divisible by both 3 and 4, which means finding numbers divisible by their LCM = 12<\/li>
  2. Identify the range boundaries for multiples of 12<\/strong>: Find the smallest and largest three-digit multiples of 12 to establish our counting range<\/li>
  3. Count the multiples using division<\/strong>: Use the range of multiples to calculate how many multiples of 12 exist in the three-digit range<\/li><\/ol>\n\n\n\n

    Execution of Strategic Approach<\/h3>\n\n\n\n

    1. Translate the problem requirements<\/h4>\n\n\n\n

    Let’s start by understanding what we’re looking for in everyday terms. We need three-digit numbers that are divisible by both 3 and 4.<\/p>\n\n\n\n

    Three-digit numbers are simply the numbers from 100 to 999 – these are all the numbers that have exactly three digits.<\/p>\n\n\n\n

    Now, when a number is divisible by both 3 and 4, it means the number can be divided evenly by both of these numbers with no remainder. For example, let’s check 12: 12 divided by<\/em> 3 = 4 (no remainder) and 12 divided by<\/em> 4 = 3 (no remainder).<\/p>\n\n\n\n

    Here’s the key insight: when we want a number divisible by both 3 and 4, we’re really looking for numbers divisible by their Least Common Multiple (LCM). Since 3 and 4 share no common factors (3 is prime and doesn’t divide 4), their LCM is simply 3 multiplied by <\/em>4 = 12.<\/p>\n\n\n\n

    So our problem becomes: How many three-digit numbers are divisible by 12?<\/strong><\/p>\n\n\n\n

    Process Skill: TRANSLATE – Converting the “divisible by both” requirement into “divisible by LCM”<\/em><\/p>\n\n\n\n

    2. Identify the range boundaries for multiples of 12<\/h4>\n\n\n\n

    Now we need to find the smallest and largest three-digit numbers that are multiples of 12.<\/p>\n\n\n\n

    Finding the smallest three-digit multiple of 12:<\/strong><\/p>\n\n\n\n

    The smallest three-digit number is 100. Let’s see: 100 divided by<\/em> 12 = 8.33…. This means 100 is between the 8th and 9th multiples of 12. Since we need the next whole multiple, we want the 9th multiple: 12 multiplied by<\/em> 9 = 108.<\/p>\n\n\n\n

    Let’s verify:  108 divided by<\/em> 3 = 36 \u2713 and  108 divided by<\/em> 4 = 27 \u2713<\/p>\n\n\n\n

    Finding the largest three-digit multiple of 12:<\/strong><\/p>\n\n\n\n

    The largest three-digit number is 999. Let’s see: 999 divided by<\/em> 12 = 83.25. This means 999 is between the 83rd and 84th multiples of 12. Since we need a whole multiple, we want the 83rd multiple: 12 multiplied by<\/em> 83 = 996.<\/p>\n\n\n\n

    Let’s verify:  996 divided by<\/em> 3 = 332 \u2713 and  996 divided by<\/em> 4 = 249 \u2713<\/p>\n\n\n\n

    So our three-digit multiples of 12 range from the 9th multiple (108) to the 83rd multiple (996).<\/p>\n\n\n\n

    3. Count the multiples using division<\/h4>\n\n\n\n

    Now we need to count how many multiples of 12 exist from the 9th multiple to the 83rd multiple, inclusive.<\/p>\n\n\n\n

    This is like counting from 9 to 83. When we count consecutive integers from a starting number to an ending number, the formula is:<\/p>\n\n\n\n

    Count = Last number – First number + 1<\/strong><\/p>\n\n\n\n

    In our case:<\/p>\n\n\n\n

    Count = 83 – 9 + 1 = 75<\/p>\n\n\n\n

    Let’s double-check this makes sense: if we had multiples from the 9th to the 11th, that would be 3 multiples (9th, 10th, 11th), and indeed  11 – 9 + 1 = 3 \u2713<\/p>\n\n\n\n

    4. Final Answer<\/h4>\n\n\n\n

    There are 75<\/strong> positive three-digit integers that are divisible by both 3 and 4.<\/p>\n\n\n\n

    This matches answer choice A. 75<\/strong><\/p>\n\n\n\n

    Quick verification:<\/strong> Our range includes multiples from 108 (12 multiplied by<\/em> 9) to 996 (12 multiplied by<\/em> 83), giving us 83 – 9 + 1 = 75 numbers total.<\/p>\n\n\n\n

    Common Faltering Points<\/h3>\n\n\n\n

    Errors while devising the approach<\/h4>\n\n\n\n
    • Misunderstanding “divisible by both”:<\/strong> Students may try to count numbers divisible by 3 and numbers divisible by 4 separately, then add them together. This double-counts numbers and ignores that we need numbers divisible by BOTH conditions simultaneously. The correct approach requires finding the LCM (12) first.<\/li>
    • Forgetting the three-digit constraint:<\/strong> Students may calculate all multiples of 12 without restricting to the range 100-999, leading to an incorrect count that includes one-digit, two-digit, or four-digit numbers.<\/li>
    • Attempting to list all numbers individually:<\/strong> Instead of using the systematic LCM approach, students may try to manually check each three-digit number for divisibility by both 3 and 4, which is time-consuming and error-prone on the GMAT.<\/li><\/ul>\n\n\n\n

      Errors while executing the approach<\/h4>\n\n\n\n
      • Incorrect boundary identification:<\/strong> When finding 100 divided by<\/em> 12 = 8.33, students may incorrectly use the 8th multiple (96) instead of rounding up to the 9th multiple (108). Similarly, for 999 divided by<\/em> 12 = 83.25, they may round up to the 84th multiple instead of down to the 83rd multiple.<\/li>
      • Arithmetic errors in multiplication:<\/strong> Students may make calculation mistakes when computing 12 multiplied by<\/em> 9 = 108 or 12 multiplied by<\/em> 83 = 996, leading to wrong boundary values and ultimately an incorrect count.<\/li>
      • Forgetting the “+1” in counting:<\/strong> When counting from the 9th to 83rd multiple, students often forget that counting consecutive integers requires the formula (last – first + 1), calculating 83 – 9 = 74 instead of 83 – 9 + 1 = 75.<\/li><\/ul>\n\n\n\n

        Errors while selecting the answer<\/h4>\n\n\n\n

        No likely faltering points – the calculation directly yields 75, which clearly matches answer choice A.<\/p>\n","protected":false},"excerpt":{"rendered":"

        How many positive three-digit integers are divisible by both 3 and 4? A. 75 B. 128 C. 150 D. 225 E. 300 Solution: Translate the problem requirements: We need to find three-digit numbers (100 to 999) that are divisible by both 3 and 4, which means finding numbers divisible by their LCM = 12 Identify the range boundaries […]<\/p>\n","protected":false},"author":102457,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"off","_et_pb_old_content":"","_et_gb_content_width":"","ub_ctt_via":""},"categories":[94,100],"tags":[],"featured_image_src":null,"author_info":{"display_name":"Kashish Garg","author_link":"https:\/\/e-gmat.com\/blogs\/author\/kashish\/"},"yoast_head":"\nHow many positive three-digit integers are divisible by both 3 and 4?<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How many positive three-digit integers are divisible by both 3 and 4?\" \/>\n<meta property=\"og:description\" content=\"How many positive three-digit integers are divisible by both 3 and 4? A. 75 B. 128 C. 150 D. 225 E. 300 Solution: Translate the problem requirements: We need to find three-digit numbers (100 to 999) that are divisible by both 3 and 4, which means finding numbers divisible by their LCM = 12 Identify the range boundaries […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html\" \/>\n<meta property=\"og:site_name\" content=\"e-GMAT Blog | Best GMAT blog on the planet\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/E-Gmat-499275643430980\" \/>\n<meta property=\"article:published_time\" content=\"2025-08-22T08:10:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-04T19:21:53+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@e_GMAT\" \/>\n<meta name=\"twitter:site\" content=\"@e_GMAT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kashish Garg\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Organization\",\"@id\":\"https:\/\/e-gmat.com\/blogs\/#organization\",\"name\":\"e-GMAT\",\"url\":\"https:\/\/e-gmat.com\/blogs\/\",\"sameAs\":[\"https:\/\/www.facebook.com\/E-Gmat-499275643430980\",\"https:\/\/www.linkedin.com\/in\/rajat-sadana-ba459a\/\",\"https:\/\/www.youtube.com\/user\/eGMATconcepts\",\"https:\/\/twitter.com\/e_GMAT\"],\"logo\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/e-gmat.com\/blogs\/#logo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/04\/logo-full.png\",\"contentUrl\":\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/04\/logo-full.png\",\"width\":908,\"height\":802,\"caption\":\"e-GMAT\"},\"image\":{\"@id\":\"https:\/\/e-gmat.com\/blogs\/#logo\"}},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/e-gmat.com\/blogs\/#website\",\"url\":\"https:\/\/e-gmat.com\/blogs\/\",\"name\":\"e-GMAT Blog | Best GMAT blog on the planet\",\"description\":\"Prepare for your GMAT and MBA journey\",\"publisher\":{\"@id\":\"https:\/\/e-gmat.com\/blogs\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/e-gmat.com\/blogs\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#webpage\",\"url\":\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html\",\"name\":\"How many positive three-digit integers are divisible by both 3 and 4?\",\"isPartOf\":{\"@id\":\"https:\/\/e-gmat.com\/blogs\/#website\"},\"datePublished\":\"2025-08-22T08:10:17+00:00\",\"dateModified\":\"2025-09-04T19:21:53+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Homepage\",\"item\":\"https:\/\/e-gmat.com\/blogs\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"GMAT Preparation\",\"item\":\"https:\/\/e-gmat.com\/blogs\/news\/gmat-prep\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"How many positive three-digit integers are divisible by both 3 and 4?\"}]},{\"@type\":\"Article\",\"@id\":\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#article\",\"isPartOf\":{\"@id\":\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#webpage\"},\"author\":{\"@id\":\"https:\/\/e-gmat.com\/blogs\/#\/schema\/person\/a3751e3cf22ad53be0877b8638a104f9\"},\"headline\":\"How many positive three-digit integers are divisible by both 3 and 4?\",\"datePublished\":\"2025-08-22T08:10:17+00:00\",\"dateModified\":\"2025-09-04T19:21:53+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#webpage\"},\"wordCount\":768,\"publisher\":{\"@id\":\"https:\/\/e-gmat.com\/blogs\/#organization\"},\"articleSection\":[\"GMAT Preparation\",\"Quant OG Solutions\"],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/e-gmat.com\/blogs\/#\/schema\/person\/a3751e3cf22ad53be0877b8638a104f9\",\"name\":\"Kashish Garg\",\"image\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/e-gmat.com\/blogs\/#personlogo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/34b808afef4b9fd7cc623102a85d3a2b?s=96&d=blank&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/34b808afef4b9fd7cc623102a85d3a2b?s=96&d=blank&r=g\",\"caption\":\"Kashish Garg\"},\"sameAs\":[\"https:\/\/e-gmat.com\/blogs\/\"],\"url\":\"https:\/\/e-gmat.com\/blogs\/author\/kashish\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"How many positive three-digit integers are divisible by both 3 and 4?","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:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html","og_locale":"en_US","og_type":"article","og_title":"How many positive three-digit integers are divisible by both 3 and 4?","og_description":"How many positive three-digit integers are divisible by both 3 and 4? A. 75 B. 128 C. 150 D. 225 E. 300 Solution: Translate the problem requirements: We need to find three-digit numbers (100 to 999) that are divisible by both 3 and 4, which means finding numbers divisible by their LCM = 12 Identify the range boundaries […]","og_url":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html","og_site_name":"e-GMAT Blog | Best GMAT blog on the planet","article_publisher":"https:\/\/www.facebook.com\/E-Gmat-499275643430980","article_published_time":"2025-08-22T08:10:17+00:00","article_modified_time":"2025-09-04T19:21:53+00:00","twitter_card":"summary_large_image","twitter_creator":"@e_GMAT","twitter_site":"@e_GMAT","twitter_misc":{"Written by":"Kashish Garg","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/e-gmat.com\/blogs\/#organization","name":"e-GMAT","url":"https:\/\/e-gmat.com\/blogs\/","sameAs":["https:\/\/www.facebook.com\/E-Gmat-499275643430980","https:\/\/www.linkedin.com\/in\/rajat-sadana-ba459a\/","https:\/\/www.youtube.com\/user\/eGMATconcepts","https:\/\/twitter.com\/e_GMAT"],"logo":{"@type":"ImageObject","@id":"https:\/\/e-gmat.com\/blogs\/#logo","inLanguage":"en-US","url":"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/04\/logo-full.png","contentUrl":"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2018\/04\/logo-full.png","width":908,"height":802,"caption":"e-GMAT"},"image":{"@id":"https:\/\/e-gmat.com\/blogs\/#logo"}},{"@type":"WebSite","@id":"https:\/\/e-gmat.com\/blogs\/#website","url":"https:\/\/e-gmat.com\/blogs\/","name":"e-GMAT Blog | Best GMAT blog on the planet","description":"Prepare for your GMAT and MBA journey","publisher":{"@id":"https:\/\/e-gmat.com\/blogs\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/e-gmat.com\/blogs\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#webpage","url":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html","name":"How many positive three-digit integers are divisible by both 3 and 4?","isPartOf":{"@id":"https:\/\/e-gmat.com\/blogs\/#website"},"datePublished":"2025-08-22T08:10:17+00:00","dateModified":"2025-09-04T19:21:53+00:00","breadcrumb":{"@id":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Homepage","item":"https:\/\/e-gmat.com\/blogs\/"},{"@type":"ListItem","position":2,"name":"GMAT Preparation","item":"https:\/\/e-gmat.com\/blogs\/news\/gmat-prep\/"},{"@type":"ListItem","position":3,"name":"How many positive three-digit integers are divisible by both 3 and 4?"}]},{"@type":"Article","@id":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#article","isPartOf":{"@id":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#webpage"},"author":{"@id":"https:\/\/e-gmat.com\/blogs\/#\/schema\/person\/a3751e3cf22ad53be0877b8638a104f9"},"headline":"How many positive three-digit integers are divisible by both 3 and 4?","datePublished":"2025-08-22T08:10:17+00:00","dateModified":"2025-09-04T19:21:53+00:00","mainEntityOfPage":{"@id":"https:\/\/neuron.e-gmat.com\/quant\/questions\/how-many-positive-three-digit-integers-are-divisible-by-both-1003.html#webpage"},"wordCount":768,"publisher":{"@id":"https:\/\/e-gmat.com\/blogs\/#organization"},"articleSection":["GMAT Preparation","Quant OG Solutions"],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/e-gmat.com\/blogs\/#\/schema\/person\/a3751e3cf22ad53be0877b8638a104f9","name":"Kashish Garg","image":{"@type":"ImageObject","@id":"https:\/\/e-gmat.com\/blogs\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/34b808afef4b9fd7cc623102a85d3a2b?s=96&d=blank&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/34b808afef4b9fd7cc623102a85d3a2b?s=96&d=blank&r=g","caption":"Kashish Garg"},"sameAs":["https:\/\/e-gmat.com\/blogs\/"],"url":"https:\/\/e-gmat.com\/blogs\/author\/kashish\/"}]}},"_links":{"self":[{"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/posts\/57463"}],"collection":[{"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/users\/102457"}],"replies":[{"embeddable":true,"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/comments?post=57463"}],"version-history":[{"count":2,"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/posts\/57463\/revisions"}],"predecessor-version":[{"id":58051,"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/posts\/57463\/revisions\/58051"}],"wp:attachment":[{"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/media?parent=57463"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/categories?post=57463"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/e-gmat.com\/blogs\/wp-json\/wp\/v2\/tags?post=57463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}