{"id":233235,"date":"2025-06-12T20:44:10","date_gmt":"2025-06-12T20:44:10","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233235"},"modified":"2025-06-12T20:44:13","modified_gmt":"2025-06-12T20:44:13","slug":"the-lcm-of-5490-and-a-third-number-is-1890-and-their-hcf-is-18","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/12\/the-lcm-of-5490-and-a-third-number-is-1890-and-their-hcf-is-18\/","title":{"rendered":"The LCM of 54,90 and a third number is 1890 and their HCF is 18 ."},"content":{"rendered":"\n<p>The LCM of 54,90 and a third number is 1890 and their HCF is 18 . what is the third number<\/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 find the third number, let\u2019s denote the three numbers as:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A = 54<\/li>\n\n\n\n<li>B = 90<\/li>\n\n\n\n<li>C = the unknown number<\/li>\n<\/ul>\n\n\n\n<p>We are told:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>LCM(54, 90, C) = 1890<\/strong><\/li>\n\n\n\n<li><strong>HCF(54, 90, C) = 18<\/strong><\/li>\n<\/ul>\n\n\n\n<p>We can use the relationship between the product of numbers, their LCM, and HCF:LCM(A,B,C)\u00d7HCF(A,B,C)=LCM(A,B)\u00d7LCM(C,HCF(A,B))\\text{LCM}(A, B, C) \\times \\text{HCF}(A, B, C) = \\text{LCM}(A, B) \\times \\text{LCM}(C, \\text{HCF}(A, B))LCM(A,B,C)\u00d7HCF(A,B,C)=LCM(A,B)\u00d7LCM(C,HCF(A,B))<\/p>\n\n\n\n<p>But an easier method here is to proceed step-by-step.<\/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: Find LCM and HCF of 54 and 90<\/h3>\n\n\n\n<p>We start by prime factorizing both numbers:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>54=2\u00d73354 = 2 \\times 3^354=2\u00d733<\/li>\n\n\n\n<li>90=2\u00d732\u00d7590 = 2 \\times 3^2 \\times 590=2\u00d732\u00d75<\/li>\n<\/ul>\n\n\n\n<p>So,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>HCF(54, 90) = 2 \u00d7 3\u00b2 = 18<\/strong><\/li>\n\n\n\n<li><strong>LCM(54, 90) = 2 \u00d7 3\u00b3 \u00d7 5 = 270<\/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\">Step 2: Use LCM of all three numbers<\/h3>\n\n\n\n<p>We are told that:LCM(54,90,C)=1890\\text{LCM}(54, 90, C) = 1890LCM(54,90,C)=1890<\/p>\n\n\n\n<p>We already know that:LCM(54,90)=270\\text{LCM}(54, 90) = 270LCM(54,90)=270<\/p>\n\n\n\n<p>So we want:LCM(270,C)=1890\\text{LCM}(270, C) = 1890LCM(270,C)=1890<\/p>\n\n\n\n<p>We now find a number C such that:LCM(270,C)=1890\\text{LCM}(270, C) = 1890LCM(270,C)=1890<\/p>\n\n\n\n<p>We use the formula:LCM(a,b)=a\u00d7bHCF(a,b)\\text{LCM}(a, b) = \\frac{a \\times b}{\\text{HCF}(a, b)}LCM(a,b)=HCF(a,b)a\u00d7b\u200b<\/p>\n\n\n\n<p>Let\u2019s solve:270\u00d7CHCF(270,C)=1890\\frac{270 \\times C}{\\text{HCF}(270, C)} = 1890HCF(270,C)270\u00d7C\u200b=1890270\u00d7C=1890\u00d7HCF(270,C)270 \\times C = 1890 \\times \\text{HCF}(270, C)270\u00d7C=1890\u00d7HCF(270,C)<\/p>\n\n\n\n<p>Now divide both sides by 270:C=1890\u00d7HCF(270,C)270C = \\frac{1890 \\times \\text{HCF}(270, C)}{270}C=2701890\u00d7HCF(270,C)\u200b<\/p>\n\n\n\n<p>Let\u2019s try HCF(270, C) = 18 (we are told that HCF of all three numbers is 18):C=1890\u00d718270=34020270=126C = \\frac{1890 \\times 18}{270} = \\frac{34020}{270} = 126C=2701890\u00d718\u200b=27034020\u200b=126<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer: The third number is <strong>126<\/strong><\/h3>\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 find the third number when the least common multiple (LCM) of three numbers is 1890 and their highest common factor (HCF) is 18, we use number theory concepts involving prime factorization and properties of LCM and HCF.<\/p>\n\n\n\n<p>Let\u2019s denote the three numbers as 54, 90, and an unknown number C. We start by finding the LCM of 54 and 90. Using prime factorization, 54 equals 2\u00d7332 \\times 3^32\u00d733, and 90 equals 2\u00d732\u00d752 \\times 3^2 \\times 52\u00d732\u00d75. The LCM is the product of the highest powers of all primes involved: 2\u00d733\u00d75=2702 \\times 3^3 \\times 5 = 2702\u00d733\u00d75=270. Their HCF is the product of the lowest powers of common primes: 2\u00d732=182 \\times 3^2 = 182\u00d732=18.<\/p>\n\n\n\n<p>Now we are told that the LCM of all three numbers is 1890. Since the LCM of 54 and 90 is already 270, we need to find a number C such that the LCM of 270 and C is 1890. We use the identity:LCM(a,b)=a\u00d7bHCF(a,b)\\text{LCM}(a, b) = \\frac{a \\times b}{\\text{HCF}(a, b)}LCM(a,b)=HCF(a,b)a\u00d7b\u200b<\/p>\n\n\n\n<p>Substituting a=270a = 270a=270, LCM=1890LCM = 1890LCM=1890, and assuming the HCF is 18 (as given), we solve for C:C=1890\u00d718270=126C = \\frac{1890 \\times 18}{270} = 126C=2701890\u00d718\u200b=126<\/p>\n\n\n\n<p>Finally, we verify that the HCF of 54, 90, and 126 is 18, and the LCM is indeed 1890, confirming our result. Therefore, the third number is <strong>126<\/strong>.<\/p>\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-banner5-427.jpeg\" alt=\"\" class=\"wp-image-233236\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The LCM of 54,90 and a third number is 1890 and their HCF is 18 . what is the third number The Correct Answer and Explanation is: To find the third number, let\u2019s denote the three numbers as: We are told: We can use the relationship between the product of numbers, their LCM, and HCF:LCM(A,B,C)\u00d7HCF(A,B,C)=LCM(A,B)\u00d7LCM(C,HCF(A,B))\\text{LCM}(A, [&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-233235","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233235","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=233235"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233235\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}