{"id":47548,"date":"2023-07-24T00:04:00","date_gmt":"2023-07-23T18:34:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=47548"},"modified":"2023-12-13T10:25:33","modified_gmt":"2023-12-13T04:55:33","slug":"rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/","title":{"rendered":"RS Aggarwal Solutions Class 10 Maths Chapter 1 &#8211; Real Numbers (Updated For 2024)"},"content":{"rendered":"\n<p><img class=\"alignnone wp-image-135354 size-full\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/07\/RS-Aggarwal-Solutions-Class-10-Maths-Chapter-1.jpg\" alt=\"RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/07\/RS-Aggarwal-Solutions-Class-10-Maths-Chapter-1.jpg 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/07\/RS-Aggarwal-Solutions-Class-10-Maths-Chapter-1-768x432.jpg 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers<\/strong>: Appearing for the Class 10 Maths Exam? You must prepare from the best help book and here we have RS Aggarwal Solutions Class 10 Maths. All the solutions are prepared by the subject matter experts. You can download the free PDF of <a href=\"https:\/\/www.kopykitab.com\/blog\/cbse-class-10-maths-rs-aggarwal-solutions\/\" target=\"_blank\" rel=\"noopener\" data-cke-saved-href=\"https:\/\/www.kopykitab.com\/blog\/cbse-class-10-maths-rs-aggarwal-solutions\/\">RS Aggarwal Solutions Class 10 Maths<\/a> Chapter 1 from here.<\/p>\n<p>Download the Free PDF of RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers using the download link given in this blog. To know more, read the whole blog.\u00a0<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"ez-toc-toggle-icon-1\"><label for=\"item-69d06e116d039\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" 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Real Numbers PDF<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/#access-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\" title=\"Access RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers\">Access RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/#access-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers-other-exercise\" title=\"Access RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers Other Exercise\u00a0\">Access RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers Other Exercise\u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/#faqs-on-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\" title=\"FAQs on RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers\">FAQs on RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/#from-where-can-i-find-the-download-link-for-the-rs-aggarwal-solutions-class-10-maths-chapter-1-pdf\" title=\"From where can I find the download link for the RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF?\">From where can I find the download link for the RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/#how-much-does-it-cost-to-download-the-rs-aggarwal-solutions-class-10-maths-chapter-1-pdf\" title=\"How much does it cost to download the\u00a0RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF?\">How much does it cost to download the\u00a0RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/#can-i-access-the-rs-aggarwal-solutions-class-10-maths-chapter-1-pdf-offline\" title=\"Can I access the\u00a0RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF offline?\">Can I access the\u00a0RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF offline?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/#how-many-exercises-are-there-in-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\" title=\"How many exercises are there in RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers?\">How many exercises are there in RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers-pdf\"><\/span>Download RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/10\/RS-Aggarwal-Solutions-Class-10-Maths-Chapter-1.pdf\" target=\"_blank\" rel=\"noopener\">RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers<\/a><\/p>\n<div id=\"example1\" style=\"text-align: justify;\">\u00a0<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 500px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br \/><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/10\/RS-Aggarwal-Solutions-Class-10-Maths-Chapter-1.pdf\", \"#example1\");<\/script><\/p>\n<h2><span class=\"ez-toc-section\" id=\"access-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\"><\/span>Access RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Question 1.<\/strong><br \/><strong>Solution:<\/strong><br \/>For any two given positive integers a and b, there exist unique whole numbers q and r such that<br \/>a = bq + r, when 0 \u2264 r &lt; b.<br \/>Here, a is called the dividend, b is a divisor, q is the quotient, and r is the remainder.<br \/>Dividend = (Divisor x Quotient) + Remainder.<\/p>\n<p><strong>Question 2.<\/strong><br \/><strong>Solution:<\/strong><br \/>Using Euclid\u2019s division Lemma<br \/>Dividend = (Divisor x Quotient) + Remainder<br \/>= (61 x 27) + 32<br \/>= 1647 + 32<br \/>= 1679<br \/>Required number = 1679<\/p>\n<p><strong>Question 3.<\/strong><br \/><strong>Solution:<\/strong><br \/>Let the required divisor = x<br \/>Then by Euclid\u2019s division Lemma,<br \/>Dividend = (Divisor x Quotient) + remainder<br \/>1365 = x x 31 + 32<br \/>=&gt; 1365 = 31x + 32<br \/>=&gt; 31x= 1365 \u2013 32 = 1333<br \/>x =\u00a0<span id=\"MathJax-Element-1-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-1\" class=\"math\"><span id=\"MathJax-Span-2\" class=\"mrow\"><span id=\"MathJax-Span-3\" class=\"mfrac\"><span id=\"MathJax-Span-4\" class=\"mn\">1331<\/span><span id=\"MathJax-Span-5\" class=\"mn\">31<\/span><\/span><\/span><\/span><\/span>\u00a0= 43<br \/>Divisor = 43<\/p>\n<p><strong>Question 4.<\/strong><br \/><strong>Solution:<\/strong><br \/>Let n be a given positive odd integer.<br \/>On dividing n by 6, let m be the quotient and r be the remainder, then by Euclid\u2019s division Lemma.<br \/>n = 6m + r, where 0 \u2264 r &lt; 6 =&gt; n = 6m + r, where r = 0, 1, 2, 3, 4, 5<br \/>=&gt; n = 6m or (6m + 1) or (6m + 2) or (6m + 3) or (6m + 4) or (6m + 5)<br \/>But n = 6m, (6m + 2) and (6m + 4) are even.<br \/>Thus when n is odd, it will be in the form of (6m + 1) (6m + 3), or (6m + 5) for some integer m.<\/p>\n<p><strong>Question 5.<\/strong><br \/><strong>Solution:<\/strong><br \/>Let n be an arbitrary odd positive integer.<br \/>On dividing by 4, let m be the quotient and r be the remainder.<br \/>So to Euclid\u2019s division lemma,<br \/>n = 4m + r, where 0 \u2264 r &lt; 4<br \/>n = 4m or (4m + 1) or (4m + 2) or (4m + 3)<br \/>But 4m and (4m + 2) are even integers.<br \/>Since n is odd, so n \u2260 4m or n \u2260 (4m + 2)<br \/>n = (4m + 1) or (4m + 3) for some integer m.<br \/>Hence any positive odd integer is of the form (4m + 1) or (4m + 3) for some integer m.<\/p>\n<p><strong>Question 6.<\/strong><br \/><strong>Solution:<\/strong><br \/>Let a = n<sup>3<\/sup>\u00a0\u2013 n<br \/>=&gt; a = n (n<sup>2<\/sup>\u00a0\u2013 1)<br \/>=&gt; a = n (n \u2013 1) (n + 1) [(a<sup>2<\/sup>\u00a0\u2013 b<sup>2<\/sup>) = (a \u2013 b) (a + b)]<br \/>=&gt; a = (n \u2013 1 ) n (n + 1)<br \/>We know that,<br \/>(i) If a number is completely divisible by 2 and 3, then it is also divisible by 6.<br \/>(ii) If the sum of digits of any number is divisible by 3, then it is also divisible by 3.<br \/>(iii) If one of the factors of any number is an even number, then it is also divisible by 2.<br \/>a = (n \u2013 1) n (n + 1) [From Eq. (i)]<br \/>Now, the sum of the digits<br \/>= n \u2013 1 + n + n + 1 = 3n<br \/>= Multiple of 3, where n is any positive integer.<br \/>and (n \u2013 1) n (n +1) will always be even, as one out of (n \u2013 1) or n or (n + 1) must be even.<br \/>Since, conditions (ii) and (iii) completely satisfy Eq. (i).<br \/>Hence, by condition (i) the number n3 \u2013 n is always divisible by 6, where n is any positive integer.<br \/>Hence proved.<\/p>\n<p><strong>Question 7.<\/strong><br \/><strong>Solution:<\/strong><br \/>Let x = 2m + 1 and y = 2m + 3 are odd positive integers, for every positive integer m.<br \/>Then, x<sup>2<\/sup>\u00a0+ y<sup>2<\/sup>\u00a0= (2m + 1)<sup>2<\/sup>\u00a0+ (2m + 3)<sup>2<\/sup><br \/>= 4m<sup>2<\/sup>\u00a0+ 1 + 4 m + 4m<sup>2<\/sup>\u00a0+ 9 + 12m [(a + b)<sup>2<\/sup>\u00a0= a<sup>2<\/sup>\u00a0+ 2ab + b<sup>2<\/sup>]<br \/>= 8m<sup>2<\/sup>\u00a0+ 16m + 10 = even<br \/>= 2(4m<sup>2<\/sup>\u00a0+ 8m + 5) or 4(2m<sup>2<\/sup>\u00a0+ 4m + 2) + 1<br \/>Hence, x<sup>2<\/sup>\u00a0+ y<sup>2<\/sup>\u00a0is even for every positive integer m but not divisible by 4.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"access-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers-other-exercise\"><\/span>Access RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers Other Exercise\u00a0<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-chapter-1-class-10-maths-exercise-1-1-solutions\/\" target=\"_blank\" rel=\"noopener\">RS Aggarwal Solutions Chapter 1 Exercise 1.1<\/a><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-chapter-1-class-10-maths-exercise-1-2-solutions\/\" target=\"_blank\" rel=\"noopener\">RS Aggarwal Solutions Chapter 1 Exercise 1.2<\/a><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-chapter-1-class-10-maths-exercise-1-3-solutions\/\" target=\"_blank\" rel=\"noopener\">RS Aggarwal Solutions Chapter 1 Exercise 1.3<\/a><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-1-exercise-1-4\/\" target=\"_blank\" rel=\"noopener\">RS Aggarwal Solutions Chapter 1 Exercise 1.4<\/a><\/p>\n<p>This is the complete blog on the RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers. To know more about the <a href=\"https:\/\/www.cbse.gov.in\/\" target=\"_blank\" rel=\"noopener\">CBSE<\/a> Class 10 Maths exam, ask in the comments.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\"><\/span>FAQs on RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1625837321576\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"from-where-can-i-find-the-download-link-for-the-rs-aggarwal-solutions-class-10-maths-chapter-1-pdf\"><\/span>From where can I find the download link for the RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can find the download link of the RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF in the above blog.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1625837356287\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"how-much-does-it-cost-to-download-the-rs-aggarwal-solutions-class-10-maths-chapter-1-pdf\"><\/span>How much does it cost to download the\u00a0RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF for free.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1633502449981\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"can-i-access-the-rs-aggarwal-solutions-class-10-maths-chapter-1-pdf-offline\"><\/span>Can I access the\u00a0RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF offline?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Once you have downloaded the RS Aggarwal Solutions Class 10 Maths Chapter 1 PDF online, you can access it offline whenever you want.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1635062941658\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"how-many-exercises-are-there-in-rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\"><\/span>How many exercises are there in RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>There are 4 exercises in this chapter.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RS Aggarwal Solutions Class 10 Maths Chapter 1 Real Numbers: Appearing for the Class 10 Maths Exam? You must prepare from the best help book and here we have RS Aggarwal Solutions Class 10 Maths. All the solutions are prepared by the subject matter experts. You can download the free PDF of RS Aggarwal Solutions &#8230; <a title=\"RS Aggarwal Solutions Class 10 Maths Chapter 1 &#8211; Real Numbers (Updated For 2024)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rs-aggarwal-solutions-class-10-maths-chapter-1-real-numbers\/\" aria-label=\"More on RS Aggarwal Solutions Class 10 Maths Chapter 1 &#8211; Real Numbers (Updated For 2024)\">Read more<\/a><\/p>\n","protected":false},"author":243,"featured_media":135354,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73412,2985,73410],"tags":[76296,77207],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/47548"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/243"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=47548"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/47548\/revisions"}],"predecessor-version":[{"id":521474,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/47548\/revisions\/521474"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/135354"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=47548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=47548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=47548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}