{"id":39144,"date":"2025-06-27T02:47:32","date_gmt":"2025-06-27T02:47:32","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=39144"},"modified":"2025-06-27T02:47:33","modified_gmt":"2025-06-27T02:47:33","slug":"find-the-least-number-by-which-1500-is-to-be-multiplied-so-that-the-product-becomes-a-perfect-square","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-least-number-by-which-1500-is-to-be-multiplied-so-that-the-product-becomes-a-perfect-square\/","title":{"rendered":"Find the least number by which 1500 is to be multiplied so that the product becomes a perfect square"},"content":{"rendered":"\n<p>Find the least number by which 1500 is to be multiplied so that the product becomes a perfect square Find the square root of the number obtained<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve this, we must find the <strong>least number<\/strong> by which <strong>1500<\/strong> should be <strong>multiplied<\/strong> so that the <strong>product is a perfect square<\/strong>, and then find the <strong>square root<\/strong> of the resulting perfect square.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Prime Factorization of 1500<\/h3>\n\n\n\n<p>We begin by breaking <strong>1500<\/strong> into its <strong>prime factors<\/strong>:1500=15\u00d7100=(3\u00d75)\u00d7(10\u00d710)=3\u00d75\u00d7(2\u00d75)21500 = 15 \\times 100 = (3 \\times 5) \\times (10 \\times 10) = 3 \\times 5 \\times (2 \\times 5)^21500=15\u00d7100=(3\u00d75)\u00d7(10\u00d710)=3\u00d75\u00d7(2\u00d75)2<\/p>\n\n\n\n<p>Now let\u2019s write this more clearly:1500=22\u00d73\u00d7531500 = 2^2 \\times 3 \\times 5^31500=22\u00d73\u00d753<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Conditions for a Perfect Square<\/h3>\n\n\n\n<p>A <strong>perfect square<\/strong> has <strong>even powers<\/strong> of all its <strong>prime factors<\/strong>.<\/p>\n\n\n\n<p>In the factorization:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>222^222 is fine since 2 is raised to an even power.<\/li>\n\n\n\n<li>313^131 is <strong>not<\/strong> a perfect square factor \u2014 the power is odd.<\/li>\n\n\n\n<li>535^353 is also not a perfect square factor \u2014 the power is odd.<\/li>\n<\/ul>\n\n\n\n<p>To make all powers even:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Multiply by <strong>3<\/strong> to make 313^131 become 323^232<\/li>\n\n\n\n<li>Multiply by <strong>5<\/strong> to make 535^353 become 545^454<\/li>\n<\/ul>\n\n\n\n<p>So, the least number to multiply by is:3\u00d75=153 \\times 5 = 153\u00d75=15<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Multiply and Find the Product<\/h3>\n\n\n\n<p>1500\u00d715=225001500 \\times 15 = 225001500\u00d715=22500<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Find the Square Root of the Result<\/h3>\n\n\n\n<p>Let\u2019s find:22500\\sqrt{22500}22500\u200b<\/p>\n\n\n\n<p>We simplify:22500=225\u00d7100=225\u00d7100=15\u00d710=150\\sqrt{22500} = \\sqrt{225 \\times 100} = \\sqrt{225} \\times \\sqrt{100} = 15 \\times 10 = 15022500\u200b=225\u00d7100\u200b=225\u200b\u00d7100\u200b=15\u00d710=150<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Least number to multiply<\/strong>: <strong>15<\/strong><\/li>\n\n\n\n<li><strong>Perfect square obtained<\/strong>: <strong>22500<\/strong><\/li>\n\n\n\n<li><strong>Square root<\/strong>: <strong>150<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation <\/h3>\n\n\n\n<p>To determine the least number by which 1500 must be multiplied to become a perfect square, we use the principle of prime factorization. A perfect square has even powers of all its prime factors. We start by factorizing 1500 into prime components. We find that 1500 equals 22\u00d73\u00d7532^2 \\times 3 \\times 5^322\u00d73\u00d753. Here, the exponent of 2 is even, but the exponents of 3 and 5 are odd. For the product to be a perfect square, all prime exponents must be even.<\/p>\n\n\n\n<p>To achieve this, we must supply enough of each prime to raise the exponents to even numbers. For the 3, we need one more factor of 3 to make it 323^232. For the 5, which is currently 535^353, we need another factor of 5 to raise it to 545^454. Therefore, we need to multiply 1500 by 3\u00d75=153 \\times 5 = 153\u00d75=15.<\/p>\n\n\n\n<p>Multiplying 1500 by 15 gives us 22500. This number has the prime factorization 22\u00d732\u00d7542^2 \\times 3^2 \\times 5^422\u00d732\u00d754, where all exponents are now even. This confirms that 22500 is indeed a perfect square.<\/p>\n\n\n\n<p>To find the square root, we take the square root of 22500. Since 22500 is 225\u00d7100225 \\times 100225\u00d7100, we take the square root of each: the square root of 225 is 15 and the square root of 100 is 10. So the final square root is 15\u00d710=15015 \\times 10 = 15015\u00d710=150.<\/p>\n\n\n\n<p>This problem demonstrates how prime factorization can be used to systematically determine the smallest multiplier needed to create a perfect square and how to efficiently compute the square root of that result.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1125.jpeg\" alt=\"\" class=\"wp-image-39145\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1125.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1125-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1125-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the least number by which 1500 is to be multiplied so that the product becomes a perfect square Find the square root of the number obtained The Correct Answer and Explanation is: To solve this, we must find the least number by which 1500 should be multiplied so that the product is a perfect [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-39144","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39144","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=39144"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39144\/revisions"}],"predecessor-version":[{"id":39146,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/39144\/revisions\/39146"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=39144"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=39144"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=39144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}