{"id":231725,"date":"2025-06-11T10:45:34","date_gmt":"2025-06-11T10:45:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=231725"},"modified":"2025-06-11T10:45:36","modified_gmt":"2025-06-11T10:45:36","slug":"v-2-find-the-smallest-number-by-which-1100-must-be-multiplied-so-that-the-product-becomes-a-perfect-square","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/11\/v-2-find-the-smallest-number-by-which-1100-must-be-multiplied-so-that-the-product-becomes-a-perfect-square\/","title":{"rendered":"V.2 Find the smallest number by which 1100 must be multiplied so that the product becomes a perfect square."},"content":{"rendered":"\n<p>V.2 Find the smallest number by which 1100 must be multiplied so that the product becomes a perfect square. Also, find the square root of the perfect square so obtained.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve the problem:<\/p>\n\n\n\n<p><strong>V.2 Find the smallest number by which 1100 must be multiplied so that the product becomes a perfect square. Also, find the square root of the perfect square so obtained.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Correct Answer:<\/strong><\/h3>\n\n\n\n<p><strong>Smallest number = 11<\/strong><br><strong>Perfect square = 1100 \u00d7 11 = 12100<\/strong><br><strong>Square root = \u221a12100 = 110<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udcd8 <strong>Explanation (Approx. 300 words):<\/strong><\/h3>\n\n\n\n<p>To make a number into a perfect square, we need to ensure that all the prime factors of the number appear in even powers. Let us begin by performing the <strong>prime factorization of 1100<\/strong>.<\/p>\n\n\n\n<p><strong>Step 1: Prime factorization of 1100<\/strong>1100=11\u00d7100=11\u00d7(10\u00d710)=11\u00d7(2\u00d75)\u00d7(2\u00d75)=11\u00d722\u00d7521100 = 11 \u00d7 100 = 11 \u00d7 (10 \u00d7 10) = 11 \u00d7 (2 \u00d7 5) \u00d7 (2 \u00d7 5) = 11 \u00d7 2^2 \u00d7 5^21100=11\u00d7100=11\u00d7(10\u00d710)=11\u00d7(2\u00d75)\u00d7(2\u00d75)=11\u00d722\u00d752<\/p>\n\n\n\n<p>So,1100=22\u00d752\u00d71111100 = 2^2 \u00d7 5^2 \u00d7 11^11100=22\u00d752\u00d7111<\/p>\n\n\n\n<p>Now, we observe the powers of the prime factors:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>222^222: Even power \u2013 this is fine.<\/li>\n\n\n\n<li>525^252: Even power \u2013 this is fine.<\/li>\n\n\n\n<li>11111^1111: Odd power \u2013 this needs to be made even.<\/li>\n<\/ul>\n\n\n\n<p>To make the power of 11 even, we must multiply the number by <strong>another 11<\/strong>. This will give:1100\u00d711=(22\u00d752\u00d7111)\u00d711=22\u00d752\u00d71121100 \u00d7 11 = (2^2 \u00d7 5^2 \u00d7 11^1) \u00d7 11 = 2^2 \u00d7 5^2 \u00d7 11^21100\u00d711=(22\u00d752\u00d7111)\u00d711=22\u00d752\u00d7112<\/p>\n\n\n\n<p>Now, all prime factors have even powers, so the product is a perfect square.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Step 2: Find the square root of the resulting perfect square<\/strong><\/p>\n\n\n\n<p>We now compute:1100\u00d711=121001100 \u00d7 11 = 121001100\u00d711=12100<\/p>\n\n\n\n<p>Take the square root of 12100:12100=22\u00d752\u00d7112=2\u00d75\u00d711=110\\sqrt{12100} = \\sqrt{2^2 \u00d7 5^2 \u00d7 11^2} = 2 \u00d7 5 \u00d7 11 = 11012100\u200b=22\u00d752\u00d7112\u200b=2\u00d75\u00d711=110<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udfe9 Final Answer:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Smallest number to multiply by = 11<\/strong><\/li>\n\n\n\n<li><strong>Perfect square = 12100<\/strong><\/li>\n\n\n\n<li><strong>Square root = 110<\/strong><\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-957.jpeg\" alt=\"\" class=\"wp-image-231726\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>V.2 Find the smallest number by which 1100 must be multiplied so that the product becomes a perfect square. Also, find the square root of the perfect square so obtained. The Correct Answer and Explanation is: To solve the problem: V.2 Find the smallest number by which 1100 must be multiplied so that the product [&hellip;]<\/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-231725","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231725","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=231725"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231725\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231725"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231725"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}