{"id":1471,"date":"2026-05-16T17:25:22","date_gmt":"2026-05-16T17:25:22","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1471"},"modified":"2026-05-16T17:26:28","modified_gmt":"2026-05-16T17:26:28","slug":"class-10-maths-real-numbers-exercise-1-1-solutions","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/class-10-maths-real-numbers-exercise-1-1-solutions\/","title":{"rendered":"Class 10 Maths Real Numbers Exercise 1.1 Solutions"},"content":{"rendered":"\t\t
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NCERT Solutions \u2013 CBSE Class 10 Mathematics<\/h2>\n\n\n\n

In this post, we have provided complete step-by-step solutions for Exercise 1.1 Real Numbers<\/strong> from the NCERT Class 10 Maths textbook. All questions are solved in an easy and exam-oriented format for better understanding.<\/p>\n\n\n\n

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Quick Information<\/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>1.1<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

Concepts Used<\/h1>\n\n\n\n
    \n
  • Prime Factorisation<\/li>\n\n\n\n
  • HCF & LCM<\/li>\n\n\n\n
  • Fundamental Theorem of Arithmetic<\/li>\n\n\n\n
  • Divisibility Rules<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Important Formula<\/h1>\n\n\n\n

    HCF<\/mtext>(<\/mo>a<\/mi>,<\/mo>b<\/mi>)<\/mo>\u00d7<\/mo>LCM<\/mtext>(<\/mo>a<\/mi>,<\/mo>b<\/mi>)<\/mo>=<\/mo>a<\/mi>\u00d7<\/mo>b<\/mi><\/mrow>\\text{HCF}(a,b) \\times \\text{LCM}(a,b)=a\\times b<\/annotation><\/semantics><\/math>HCF(a,b)\u00d7LCM(a,b)=a\u00d7b<\/p>\n\n\n\n
    \n\n\n\n

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

    Express each number as a product of its prime factors.<\/h2>\n\n\n\n
    \n\n\n\n

    (i) 140<\/h2>\n\n\n\n

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

    Prime factorisation of 140:<\/p>\n\n\n\n

    140 = 2 \u00d7 70
    70 = 2 \u00d7 35
    35 = 5 \u00d7 7<\/code><\/pre>\n\n\n\n
    Therefore,<\/p>\n\n\n\n
    140 = 2 \u00d7 2 \u00d7 5 \u00d7 7<\/code><\/pre>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    140 = 2\u00b2 \u00d7 5 \u00d7 7<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    (ii) 156<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    156 = 2 \u00d7 78
    78 = 2 \u00d7 39
    39 = 3 \u00d7 13<\/code><\/pre>\n\n\n\n
    Therefore,<\/p>\n\n\n\n
    156 = 2 \u00d7 2 \u00d7 3 \u00d7 13<\/code><\/pre>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    156 = 2\u00b2 \u00d7 3 \u00d7 13<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    (iii) 3825<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    3825 = 5 \u00d7 765
    765 = 5 \u00d7 153
    153 = 3 \u00d7 51
    51 = 3 \u00d7 17<\/code><\/pre>\n\n\n\n
    Therefore,<\/p>\n\n\n\n
    3825 = 5 \u00d7 5 \u00d7 3 \u00d7 3 \u00d7 17<\/code><\/pre>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    3825 = 3\u00b2 \u00d7 5\u00b2 \u00d7 17<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    (iv) 5005<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    5005 = 5 \u00d7 1001
    1001 = 7 \u00d7 143
    143 = 11 \u00d7 13<\/code><\/pre>\n\n\n\n
    Therefore,<\/p>\n\n\n\n
    5005 = 5 \u00d7 7 \u00d7 11 \u00d7 13<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    (v) 7429<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    7429 = 17 \u00d7 437
    437 = 19 \u00d7 23<\/code><\/pre>\n\n\n\n
    Therefore,<\/p>\n\n\n\n
    7429 = 17 \u00d7 19 \u00d7 23<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 2<\/h1>\n\n\n\n
    Find the LCM and HCF of the following pairs of integers and verify that:<\/h2>\n\n\n\n
    LCM<\/mtext>\u00d7<\/mo>HCF<\/mtext>=<\/mo>Product of the two numbers<\/mtext><\/mrow>\\text{LCM} \\times \\text{HCF}=\\text{Product of the two numbers}<\/annotation><\/semantics><\/math>LCM\u00d7HCF=Product of the two numbers<\/p>\n\n\n\n
    \n\n\n\n
    (i) 26 and 91<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Prime factorisation:<\/p>\n\n\n\n
    26 = 2 \u00d7 13
    91 = 7 \u00d7 13<\/code><\/pre>\n\n\n\n
    HCF:<\/p>\n\n\n\n
    HCF = 13<\/code><\/pre>\n\n\n\n
    LCM:<\/p>\n\n\n\n
    LCM = 2 \u00d7 7 \u00d7 13 = 182<\/code><\/pre>\n\n\n\n
    Verification:<\/p>\n\n\n\n
    13 \u00d7 182 = 2366
    26 \u00d7 91 = 2366<\/code><\/pre>\n\n\n\n
    Hence verified.<\/p>\n\n\n\n
    \n\n\n\n
    (ii) 510 and 92<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Prime factorisation:<\/p>\n\n\n\n
    510 = 2 \u00d7 3 \u00d7 5 \u00d7 17
    92 = 2\u00b2 \u00d7 23<\/code><\/pre>\n\n\n\n
    HCF:<\/p>\n\n\n\n
    HCF = 2<\/code><\/pre>\n\n\n\n
    LCM:<\/p>\n\n\n\n
    LCM = 2\u00b2 \u00d7 3 \u00d7 5 \u00d7 17 \u00d7 23
    = 23460<\/code><\/pre>\n\n\n\n
    Verification:<\/p>\n\n\n\n
    23460 \u00d7 2 = 46920
    510 \u00d7 92 = 46920<\/code><\/pre>\n\n\n\n
    Hence verified.<\/p>\n\n\n\n
    \n\n\n\n
    (iii) 336 and 54<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Prime factorisation:<\/p>\n\n\n\n
    336 = 2\u2074 \u00d7 3 \u00d7 7
    54 = 2 \u00d7 3\u00b3<\/code><\/pre>\n\n\n\n
    HCF:<\/p>\n\n\n\n
    HCF = 2 \u00d7 3 = 6<\/code><\/pre>\n\n\n\n
    LCM:<\/p>\n\n\n\n
    LCM = 2\u2074 \u00d7 3\u00b3 \u00d7 7
    = 3024<\/code><\/pre>\n\n\n\n
    Verification:<\/p>\n\n\n\n
    3024 \u00d7 6 = 18144
    336 \u00d7 54 = 18144<\/code><\/pre>\n\n\n\n
    Hence verified.<\/p>\n\n\n\n
    \n\n\n\n
    Question 3<\/h1>\n\n\n\n
    Find the LCM and HCF using prime factorisation method.<\/h2>\n\n\n\n
    \n\n\n\n
    (i) 12, 15 and 21<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Prime factorisation:<\/p>\n\n\n\n
    12 = 2\u00b2 \u00d7 3
    15 = 3 \u00d7 5
    21 = 3 \u00d7 7<\/code><\/pre>\n\n\n\n
    HCF:<\/p>\n\n\n\n
    HCF = 3<\/code><\/pre>\n\n\n\n
    LCM:<\/p>\n\n\n\n
    LCM = 2\u00b2 \u00d7 3 \u00d7 5 \u00d7 7
    = 420<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    (ii) 17, 23 and 29<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    All numbers are prime.<\/p>\n\n\n\n
    HCF:<\/p>\n\n\n\n
    HCF = 1<\/code><\/pre>\n\n\n\n
    LCM:<\/p>\n\n\n\n
    LCM = 17 \u00d7 23 \u00d7 29
    = 11339<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    (iii) 8, 9 and 25<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Prime factorisation:<\/p>\n\n\n\n
    8 = 2\u00b3
    9 = 3\u00b2
    25 = 5\u00b2<\/code><\/pre>\n\n\n\n
    HCF:<\/p>\n\n\n\n
    HCF = 1<\/code><\/pre>\n\n\n\n
    LCM:<\/p>\n\n\n\n
    LCM = 2\u00b3 \u00d7 3\u00b2 \u00d7 5\u00b2
    = 1800<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 4<\/h1>\n\n\n\n
    Given that HCF(306, 657) = 9, find LCM(306, 657).<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Using formula:<\/p>\n\n\n\n
    LCM<\/mtext>=<\/mo>a<\/mi>\u00d7<\/mo>b<\/mi><\/mrow>HCF<\/mtext><\/mfrac><\/mrow>\\text{LCM}=\\frac{a\\times b}{\\text{HCF}}<\/annotation><\/semantics><\/math>LCM=HCFa\u00d7b\u200b<\/p>\n\n\n\n
    Substituting values:<\/p>\n\n\n\n
    LCM = (306 \u00d7 657) \/ 9<\/code><\/pre>\n\n\n\n
    = 22338<\/code><\/pre>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    LCM = 22338<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Question 5<\/h1>\n\n\n\n
    Check whether 6\u207f can end with digit 0 for any natural number n.<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    We know:<\/p>\n\n\n\n
    6 = 2 \u00d7 3<\/code><\/pre>\n\n\n\n
    Therefore:<\/p>\n\n\n\n
    6\u207f = 2\u207f \u00d7 3\u207f<\/code><\/pre>\n\n\n\n
    A number ending with 0 must contain factor 5.<\/p>\n\n\n\n
    But prime factorisation of 6\u207f contains only 2 and 3.<\/p>\n\n\n\n
    Therefore, 6\u207f can never end with digit 0.<\/p>\n\n\n\n
    \n\n\n\n
    Question 6<\/h1>\n\n\n\n
    Explain why the following are composite numbers.<\/h2>\n\n\n\n
    (i) 7 \u00d7 11 \u00d7 13 + 13<\/h3>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Taking 13 common:<\/p>\n\n\n\n
    = 13(7 \u00d7 11 + 1)<\/code><\/pre>\n\n\n\n
    = 13(77 + 1)<\/code><\/pre>\n\n\n\n
    = 13 \u00d7 78<\/code><\/pre>\n\n\n\n
    Since it has factors other than 1 and itself, it is composite.<\/p>\n\n\n\n
    \n\n\n\n
    (ii) 7 \u00d7 6 \u00d7 5 \u00d7 4 \u00d7 3 \u00d7 2 \u00d7 1 + 5<\/h3>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    Taking 5 common:<\/p>\n\n\n\n
    = 5(7 \u00d7 6 \u00d7 4 \u00d7 3 \u00d7 2 \u00d7 1 + 1)<\/code><\/pre>\n\n\n\n
    Hence it is divisible by 5.<\/p>\n\n\n\n
    Therefore, it is composite.<\/p>\n\n\n\n
    \n\n\n\n
    Question 7<\/h1>\n\n\n\n
    Sonia takes 18 minutes and Ravi takes 12 minutes to complete one round of a sports field. After how many minutes will they meet again at the starting point?<\/h2>\n\n\n\n
    Solution<\/h3>\n\n\n\n
    We find LCM of 18 and 12.<\/p>\n\n\n\n
    Prime factorisation:<\/p>\n\n\n\n
    18 = 2 \u00d7 3\u00b2
    12 = 2\u00b2 \u00d7 3<\/code><\/pre>\n\n\n\n
    LCM:<\/p>\n\n\n\n
    LCM = 2\u00b2 \u00d7 3\u00b2
    = 36<\/code><\/pre>\n\n\n\n
    Final Answer:<\/p>\n\n\n\n
    They will meet again after 36 minutes.<\/code><\/pre>\n\n\n\n
    \n\n\n\n
    Common Mistakes<\/h1>\n\n\n\n
      \n
    • Wrong prime factorisation<\/li>\n\n\n\n
    • Forgetting highest\/smallest powers in LCM & HCF<\/li>\n\n\n\n
    • Calculation mistakes in multiplication<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n
      Exam Tips<\/h1>\n\n\n\n
        \n
      • Learn prime tables till 50<\/li>\n\n\n\n
      • Practice factorisation daily<\/li>\n\n\n\n
      • Verify LCM \u00d7 HCF relation carefully<\/li>\n<\/ul>\n\n\n\n
        \n\n\n\n
        FAQs<\/h1>\n\n\n\n
        Is Exercise 1.1 important for board exams?<\/h2>\n\n\n\n
        Yes, questions based on HCF, LCM and prime factorisation are frequently asked in CBSE exams.<\/p>\n\n\n\n
        \n\n\n\n
        Which theorem is used in this exercise?<\/h2>\n\n\n\n
        Fundamental Theorem of Arithmetic.<\/p>\n\n\n\n
        \n\n\n\n
        What is the best way to solve these questions?<\/h2>\n\n\n\n
        Use step-by-step prime factorisation carefully and verify calculations.<\/p>\n\n\n\n
        \n\n\n\n
        Related Topics<\/h1>\n\n\n\n
          \n
        • Real Numbers Important Questions<\/li>\n\n\n\n
        • Real Numbers MCQs<\/li>\n\n\n\n
        • Real Numbers Notes<\/li>\n\n\n\n
        • Exercise 1.2 Solutions<\/li>\n<\/ul>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\n    
          \n\n\t\t\n        
            \n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"
            NCERT Solutions \u2013 CBSE Class 10 Mathematics In this post, we have provided complete step-by-step solutions for Exercise<\/p>\n","protected":false},"author":1,"featured_media":1472,"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":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[22,5,10,23,1],"tags":[],"class_list":["post-1471","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cbse","category-class-10","category-maths-class-10","category-real-numbers","category-uncategorized"],"jetpack_publicize_connections":[],"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1471","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=1471"}],"version-history":[{"count":4,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1471\/revisions"}],"predecessor-version":[{"id":1476,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1471\/revisions\/1476"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/1472"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=1471"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=1471"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=1471"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}