{"id":245662,"date":"2025-07-06T10:44:28","date_gmt":"2025-07-06T10:44:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=245662"},"modified":"2025-07-06T10:44:32","modified_gmt":"2025-07-06T10:44:32","slug":"how-many-prime-numbers-less-than-100-and-greater-than-3-are-of-the-form","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/06\/how-many-prime-numbers-less-than-100-and-greater-than-3-are-of-the-form\/","title":{"rendered":"How many prime numbers less than 100 and greater than 3 are of the form"},"content":{"rendered":"\n

How many prime numbers less than 100 and greater than 3 are of the form: 4x + 1, 5y – 1?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To find how many prime numbers less than 100 and greater than 3 are of the form 4x+14x + 14x+1 and 5y\u221215y – 15y\u22121, we need to first identify the prime numbers in this range and then check which ones satisfy both equations.<\/p>\n\n\n\n

Step 1: List all prime numbers greater than 3 and less than 100<\/h3>\n\n\n\n

The prime numbers greater than 3 and less than 100 are:<\/p>\n\n\n\n

5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.<\/p>\n\n\n\n

Step 2: Determine which of these prime numbers are of the form 4x+14x + 14x+1<\/h3>\n\n\n\n

We need to check each prime number to see if it can be written as 4x+14x + 14x+1 for some integer xxx. This means the prime number must leave a remainder of 1 when divided by 4.<\/p>\n\n\n\n

Performing the division:<\/p>\n\n\n\n

    \n
  • 5\u00f74=15 \\div 4 = 15\u00f74=1 remainder 1, so 5 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 7\u00f74=17 \\div 4 = 17\u00f74=1 remainder 3, so 7 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 11\u00f74=211 \\div 4 = 211\u00f74=2 remainder 3, so 11 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 13\u00f74=313 \\div 4 = 313\u00f74=3 remainder 1, so 13 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 17\u00f74=417 \\div 4 = 417\u00f74=4 remainder 1, so 17 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 19\u00f74=419 \\div 4 = 419\u00f74=4 remainder 3, so 19 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 23\u00f74=523 \\div 4 = 523\u00f74=5 remainder 3, so 23 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 29\u00f74=729 \\div 4 = 729\u00f74=7 remainder 1, so 29 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 31\u00f74=731 \\div 4 = 731\u00f74=7 remainder 3, so 31 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 37\u00f74=937 \\div 4 = 937\u00f74=9 remainder 1, so 37 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 41\u00f74=1041 \\div 4 = 1041\u00f74=10 remainder 1, so 41 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 43\u00f74=1043 \\div 4 = 1043\u00f74=10 remainder 3, so 43 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 47\u00f74=1147 \\div 4 = 1147\u00f74=11 remainder 3, so 47 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 53\u00f74=1353 \\div 4 = 1353\u00f74=13 remainder 1, so 53 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 59\u00f74=1459 \\div 4 = 1459\u00f74=14 remainder 3, so 59 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 61\u00f74=1561 \\div 4 = 1561\u00f74=15 remainder 1, so 61 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 67\u00f74=1667 \\div 4 = 1667\u00f74=16 remainder 3, so 67 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 71\u00f74=1771 \\div 4 = 1771\u00f74=17 remainder 3, so 71 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 73\u00f74=1873 \\div 4 = 1873\u00f74=18 remainder 1, so 73 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 79\u00f74=1979 \\div 4 = 1979\u00f74=19 remainder 3, so 79 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 83\u00f74=2083 \\div 4 = 2083\u00f74=20 remainder 3, so 83 is not of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 89\u00f74=2289 \\div 4 = 2289\u00f74=22 remainder 1, so 89 is of the form 4x+14x + 14x+1.<\/li>\n\n\n\n
  • 97\u00f74=2497 \\div 4 = 2497\u00f74=24 remainder 1, so 97 is of the form 4x+14x + 14x+1.<\/li>\n<\/ul>\n\n\n\n

    So, the prime numbers less than 100 that are of the form 4x+14x + 14x+1 are:<\/p>\n\n\n\n

    5,13,17,29,37,41,53,61,73,89,97.5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97.5,13,17,29,37,41,53,61,73,89,97.<\/p>\n\n\n\n

    Step 3: Determine which of these prime numbers are of the form 5y\u221215y – 15y\u22121<\/h3>\n\n\n\n

    Next, we need to check which of these numbers are of the form 5y\u221215y – 15y\u22121, meaning they must leave a remainder of 4 when divided by 5.<\/p>\n\n\n\n

    Performing the division:<\/p>\n\n\n\n

      \n
    • 5\u00f75=15 \\div 5 = 15\u00f75=1 remainder 0, so 5 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 13\u00f75=213 \\div 5 = 213\u00f75=2 remainder 3, so 13 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 17\u00f75=317 \\div 5 = 317\u00f75=3 remainder 2, so 17 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 29\u00f75=529 \\div 5 = 529\u00f75=5 remainder 4, so 29 is of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 37\u00f75=737 \\div 5 = 737\u00f75=7 remainder 2, so 37 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 41\u00f75=841 \\div 5 = 841\u00f75=8 remainder 1, so 41 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 53\u00f75=1053 \\div 5 = 1053\u00f75=10 remainder 3, so 53 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 61\u00f75=1261 \\div 5 = 1261\u00f75=12 remainder 1, so 61 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 73\u00f75=1473 \\div 5 = 1473\u00f75=14 remainder 3, so 73 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 89\u00f75=1789 \\div 5 = 1789\u00f75=17 remainder 4, so 89 is of the form 5y\u221215y – 15y\u22121.<\/li>\n\n\n\n
    • 97\u00f75=1997 \\div 5 = 1997\u00f75=19 remainder 2, so 97 is not of the form 5y\u221215y – 15y\u22121.<\/li>\n<\/ul>\n\n\n\n

      Step 4: Final Answer<\/h3>\n\n\n\n

      The prime numbers less than 100 and greater than 3 that satisfy both 4x+14x + 14x+1 and 5y\u221215y – 15y\u22121 are:<\/p>\n\n\n\n

      29,89.29, 89.29,89.<\/p>\n\n\n\n

      Thus, there are 2 prime numbers<\/strong> that satisfy both conditions.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      How many prime numbers less than 100 and greater than 3 are of the form: 4x + 1, 5y – 1? The Correct Answer and Explanation is: To find how many prime numbers less than 100 and greater than 3 are of the form 4x+14x + 14x+1 and 5y\u221215y – 15y\u22121, we need to first […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-245662","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245662","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=245662"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245662\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245662"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245662"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245662"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}