{"id":240929,"date":"2025-07-03T13:42:04","date_gmt":"2025-07-03T13:42:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=240929"},"modified":"2025-07-03T13:42:05","modified_gmt":"2025-07-03T13:42:05","slug":"find-the-smallest-positive-integer-to-multiply-by-896-and-create-a-perfect-squared-number","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/find-the-smallest-positive-integer-to-multiply-by-896-and-create-a-perfect-squared-number\/","title":{"rendered":"Find the smallest positive integer to multiply by 896 and create a perfect squared number"},"content":{"rendered":"\n

Find the smallest positive integer to multiply by 896 and create a perfect squared number<\/p>\n\n\n\n

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

To find the smallest positive integer that, when multiplied by 896, results in a perfect square, we need to factorize 896 and analyze its prime factorization.<\/p>\n\n\n\n

Step 1: Prime Factorization of 896<\/h3>\n\n\n\n

First, we find the prime factorization of 896 by dividing it by the smallest primes:896\u00f72=448896 \\div 2 = 448896\u00f72=448448\u00f72=224448 \\div 2 = 224448\u00f72=224224\u00f72=112224 \\div 2 = 112224\u00f72=112112\u00f72=56112 \\div 2 = 56112\u00f72=5656\u00f72=2856 \\div 2 = 2856\u00f72=2828\u00f72=1428 \\div 2 = 1428\u00f72=1414\u00f72=714 \\div 2 = 714\u00f72=7<\/p>\n\n\n\n

Now we have 896=27\u00d77896 = 2^7 \\times 7896=27\u00d77.<\/p>\n\n\n\n

Step 2: Conditions for a Perfect Square<\/h3>\n\n\n\n

For a number to be a perfect square, every prime factor in its prime factorization must have an even exponent. This means that in the factorization of 896, we have:<\/p>\n\n\n\n

    \n
  • 272^727, which has an odd exponent (7), and<\/li>\n\n\n\n
  • 717^171, which also has an odd exponent (1).<\/li>\n<\/ul>\n\n\n\n

    Step 3: Making the Exponents Even<\/h3>\n\n\n\n

    To make the exponents of the primes even, we need to multiply by:<\/p>\n\n\n\n

      \n
    • One more factor of 222 to make the exponent of 2 even: 212^121, and<\/li>\n\n\n\n
    • One more factor of 777 to make the exponent of 7 even: 717^171.<\/li>\n<\/ul>\n\n\n\n

      Step 4: Finding the Multiplier<\/h3>\n\n\n\n

      So, we multiply 896 by 2\u00d77=142 \\times 7 = 142\u00d77=14. This gives:896\u00d714=28\u00d772896 \\times 14 = 2^8 \\times 7^2896\u00d714=28\u00d772<\/p>\n\n\n\n

      Now, the exponents of both prime factors are even, so the product 896\u00d714896 \\times 14896\u00d714 is a perfect square.<\/p>\n\n\n\n

      Step 5: Conclusion<\/h3>\n\n\n\n

      Thus, the smallest positive integer to multiply by 896 to make it a perfect square is 14<\/strong>.<\/p>\n\n\n\n

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

      Find the smallest positive integer to multiply by 896 and create a perfect squared number The Correct Answer and Explanation is: To find the smallest positive integer that, when multiplied by 896, results in a perfect square, we need to factorize 896 and analyze its prime factorization. Step 1: Prime Factorization of 896 First, we […]<\/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-240929","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/240929","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=240929"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/240929\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=240929"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=240929"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=240929"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}