{"id":1565,"date":"2026-05-17T16:17:17","date_gmt":"2026-05-17T16:17:17","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1565"},"modified":"2026-05-19T07:01:42","modified_gmt":"2026-05-19T07:01:42","slug":"class-10-maths-real-numbers-exercise-1-3-solutions-euclids-division-lemma-ncert-solutions","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/class-10-maths-real-numbers-exercise-1-3-solutions-euclids-division-lemma-ncert-solutions\/","title":{"rendered":"Class 10 Maths Real Numbers Exercise 1.3 Solutions | Euclid\u2019s Division Lemma NCERT Solutions"},"content":{"rendered":"\n
\n\n\n\n

Short Intro<\/h1>\n\n\n\n

In this post, students can find complete step-by-step solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.3 based on the latest NCERT syllabus. All questions are solved in a simple and exam-oriented format to help students understand Euclid\u2019s Division Lemma and related concepts easily.<\/p>\n\n\n\n

Exercise 1.3 is based on Euclid\u2019s Division Algorithm and finding HCF using Euclid\u2019s method.<\/p>\n\n\n\n

\n\n\n\n

Quick Information Box<\/h1>\n\n\n\n
Item<\/th>Details<\/th><\/tr><\/thead>
Board<\/td>CBSE<\/td><\/tr>
Class<\/td>10<\/td><\/tr>
Subject<\/td>Maths<\/td><\/tr>
Chapter<\/td>Real Numbers<\/td><\/tr>
Exercise<\/td>1.3<\/td><\/tr>
Main Topic<\/td>Euclid\u2019s Division Lemma<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

Concepts Used (Topics Covered)<\/h1>\n\n\n\n
    \n
  • Euclid\u2019s Division Lemma<\/li>\n\n\n\n
  • HCF using Euclid\u2019s Algorithm<\/li>\n\n\n\n
  • Divisibility<\/li>\n\n\n\n
  • Prime Numbers<\/li>\n\n\n\n
  • Rational and Irrational Numbers<\/li>\n<\/ul>\n\n\n\n

    The chapter explains divisibility properties and applications of Euclid\u2019s algorithm.<\/p>\n\n\n\n

    \n\n\n\n

    Important Formulas<\/h1>\n\n\n\n

    Euclid\u2019s Division Lemma<\/h2>\n\n\n\n

    a=bq+r, 0\u2264r<b<\/p>\n\n\n\n

    \n\n\n\n

    HCF Formula<\/h2>\n\n\n\n

    HCF(a,b)=HCF(b,r)<\/p>\n\n\n\n

    \n\n\n\n

    Questions & Step-by-step Solutions<\/h1>\n\n\n\n

    Question 1<\/h1>\n\n\n\n

    Use Euclid\u2019s division algorithm to find the HCF of 135 and 225.<\/h2>\n\n\n\n

    Solution<\/h3>\n\n\n\n

    Using Euclid\u2019s Division Lemma:<\/p>\n\n\n\n

    225<\/mn>=<\/mo>135<\/mn>\u00d7<\/mo>1<\/mn>+<\/mo>90<\/mn><\/mrow>225=135\\times1+90<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

    Now,<\/p>\n\n\n\n

    135<\/mn>=<\/mo>90<\/mn>\u00d7<\/mo>1<\/mn>+<\/mo>45<\/mn><\/mrow>135=90\\times1+45<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

    Again,<\/p>\n\n\n\n

    90=45\u00d72+0<\/p>\n\n\n\n

    Since remainder becomes 0,<\/p>\n\n\n\n

    Final Answer:<\/p>\n\n\n\n

    HCF = 45\n<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 2<\/h1>\n\n\n\n
    Use Euclid\u2019s division algorithm to find the HCF of 196 and 38220.<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Applying Euclid\u2019s algorithm:<\/p>\n\n\n\n
    38220=196\u00d7195+0<\/p>\n\n\n\n
    Since remainder is 0,<\/p>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    HCF = 196\n<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 3<\/h1>\n\n\n\n
    Use Euclid\u2019s division algorithm to find the HCF of 867 and 255.<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Applying Euclid\u2019s division lemma:<\/p>\n\n\n\n
    867=255\u00d73+102<\/p>\n\n\n\n
    Now,<\/p>\n\n\n\n
    255=102×2+51<\/p>\n\n\n\n
    Again,<\/p>\n\n\n\n
    102=51×2+0<\/p>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    HCF = 51\n<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 4<\/h1>\n\n\n\n
    A school has received 96 books for Class X and 404 books for Class IX. These books are to be arranged in stacks so that each stack has the same number of books. Find the maximum number of books that each stack can have.<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    We need to find:<\/p>\n\n\n\n
    HCF of 96 and 404\n<\/code><\/pre>\n\n\n\n
    Using Euclid\u2019s algorithm:<\/p>\n\n\n\n
    404=96×4+20<\/p>\n\n\n\n
    Now,<\/p>\n\n\n\n
    96=20×4+16<\/p>\n\n\n\n
    Again,<\/p>\n\n\n\n
    20=16×1+4<\/p>\n\n\n\n
    Now,<\/p>\n\n\n\n
    16=4×4+0<\/p>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    Maximum number of books in each stack = 4\n<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 5<\/h1>\n\n\n\n
    Find the largest number that divides 70 and 125 leaving remainders 5 and 8 respectively.<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Required divisor:<\/p>\n\n\n\n
    HCF of (70 \u2212 5) and (125 \u2212 8)\n<\/code><\/pre>\n\n\n\n
    = HCF of 65 and 117\n<\/code><\/pre>\n\n\n\n
    Using Euclid\u2019s algorithm:<\/p>\n\n\n\n
    117=65×1+52<\/p>\n\n\n\n
    Now,<\/p>\n\n\n\n
    65=52×1+13<\/p>\n\n\n\n
    Again,<\/p>\n\n\n\n
    52=13×4+0<\/p>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    Required number = 13\n<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 6<\/h1>\n\n\n\n
    Find whether 6\u207f can end with digit 0 for any natural number n.<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Prime factorisation of:<\/p>\n\n\n\n
    6n<\/sup>=(2X3)n<\/sup><\/p>\n\n\n\n
    A number ending with 0 must contain factor 5.<\/p>\n\n\n\n
    But:<\/p>\n\n\n\n
    6\u207f contains only factors 2 and 3.\n<\/code><\/pre>\n\n\n\n
    Therefore,<\/p>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    6\u207f can never end with digit 0.\n<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Common Mistakes<\/h1>\n\n\n\n
      \n
    • Writing remainder greater than divisor<\/li>\n\n\n\n
    • Incorrect division calculations<\/li>\n\n\n\n
    • Stopping Euclid\u2019s algorithm too early<\/li>\n\n\n\n
    • Confusing HCF and LCM<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n
      Exam Tips<\/h1>\n\n\n\n
        \n
      • Always use:<\/li>\n<\/ul>\n\n\n\n
        a=bq+r<\/p>\n\n\n\n
        properly.<\/p>\n\n\n\n
          \n
        • Continue division until remainder becomes 0.<\/li>\n\n\n\n
        • Write each step clearly for full marks.<\/li>\n\n\n\n
        • Practice mental division for faster calculations.<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n
          Practice MCQs<\/h1>\n\n\n\n
          MCQ 1<\/h2>\n\n\n\n
          Which formula represents Euclid\u2019s Division Lemma?<\/p>\n\n\n\n
          A. a = b + r
          B. a = bq + r
          C. a = br
          D. a = q + r<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          B. a = bq + r\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          MCQ 2<\/h2>\n\n\n\n
          The remainder in Euclid\u2019s lemma is always:<\/p>\n\n\n\n
          A. Greater than divisor
          B. Equal to divisor
          C. Less than divisor
          D. Negative<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          C. Less than divisor\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          MCQ 3<\/h2>\n\n\n\n
          HCF of 12 and 18 is:<\/p>\n\n\n\n
          A. 2
          B. 3
          C. 6
          D. 9<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          C. 6\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          MCQ 4<\/h2>\n\n\n\n
          Which method is used to find HCF quickly?<\/p>\n\n\n\n
          A. Graph method
          B. Euclid\u2019s algorithm
          C. Matrix method
          D. Elimination method<\/p>\n\n\n\n
          Answer:<\/p>\n\n\n\n
          B. Euclid\u2019s algorithm\n<\/code><\/pre>\n\n\n\n
          \n\n\n\n
          FAQ Section<\/h1>\n\n\n\n
          What is Euclid\u2019s Division Lemma?<\/h2>\n\n\n\n
          It states that:<\/p>\n\n\n\n
          a=bq+r<\/p>\n\n\n\n
          where:<\/p>\n\n\n\n
          0\\leq r<b<\/p>\n\n\n\n
          \n\n\n\n
          Why is Exercise 1.3 important?<\/h2>\n\n\n\n
          It helps students understand HCF calculation and divisibility concepts used in board exams.<\/p>\n\n\n\n
          \n\n\n\n
          Which chapter contains Euclid\u2019s Algorithm?<\/h2>\n\n\n\n
          Chapter 1 \u2013 Real Numbers.<\/p>\n\n\n\n
          \n\n\n\n
          Is Euclid\u2019s Algorithm important for competitive exams?<\/h2>\n\n\n\n
          Yes, it is frequently used in aptitude and number system questions.<\/p>\n\n\n\n
          \n\n\n\n
          \ud83d\udcd8 Improve Your Maths Preparation with MyMockMate!<\/p>\n\n\n\n
          \u2705 Practice Chapter-wise Mock Tests
          \u2705 Solve Important MCQs
          \u2705 Download NCERT Notes
          \u2705 Get Instant Performance Analysis
          \u2705 Prepare for Board Exams Smarter<\/p>\n\n\n\n
          Start learning now on MyMockMate.<\/p>\n\n\n\n
          \n\n\n\n
          <\/p>\n\n    
          \n\n\t\t\n        
            \n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"
            Short Intro In this post, students can find complete step-by-step solutions for Class 10 Maths Chapter 1 Real<\/p>\n","protected":false},"author":1,"featured_media":1566,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_angie_page":false,"_surecart_dashboard_logo_width":"180px","_surecart_dashboard_show_logo":true,"_surecart_dashboard_navigation_orders":true,"_surecart_dashboard_navigation_invoices":true,"_surecart_dashboard_navigation_subscriptions":true,"_surecart_dashboard_navigation_downloads":true,"_surecart_dashboard_navigation_billing":true,"_surecart_dashboard_navigation_account":true,"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"page_builder":"","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[23],"tags":[61,48,74,44,70,75,67,69,66,65,68,72,62,76,71,47,45,50,73],"class_list":["post-1565","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-real-numbers","tag-board-exam-maths","tag-cbse-class-10-maths","tag-cbse-maths-notes","tag-class-10-maths","tag-class-10-maths-solutions","tag-class-10-ncert-solutions","tag-euclid-algorithm","tag-euclid-division-algorithm-questions","tag-euclid-division-lemma","tag-exercise-1-3-solutions","tag-hcf-questions","tag-hcf-using-euclid-algorithm","tag-maths-chapter-1-solutions","tag-maths-important-questions","tag-ncert-maths-chapter-1","tag-ncert-solutions","tag-real-numbers","tag-real-numbers-chapter-1","tag-real-numbers-exercise-1-3"],"jetpack_publicize_connections":[],"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1565","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=1565"}],"version-history":[{"count":3,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1565\/revisions"}],"predecessor-version":[{"id":1586,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1565\/revisions\/1586"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/1566"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=1565"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=1565"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=1565"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}