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SubjectsMathematics
अभाज्य गुणनखंडन द्वारा महत्तम समापवर्तक और लघुत्तम समापवर्त्य
In this Class 10 Mathematics topic from Real Numbers, students learn to find the HCF and LCM of two or more numbers through prime factorisation. They break each number into prime factors, compare the resulting powers, and select the appropriate common factors: the lowest powers for HCF and the highest powers for LCM. The topic also develops accuracy in writing factor trees, simplifies comparison between methods, and helps students apply these concepts to numerical and everyday problem-solving situations.
Correct answer: B. (6)
Explanation: Step 1: (18=2\times3^2) and (24=2^3\times3). Step 2: Taking the smaller powers of common prime factors gives (2\times3=6). Step 3: For HCF, take only common prime factors.
Correct answer: C. (72)
Explanation: Step 1: (18=2\times3^2) and (24=2^3\times3). Step 2: Taking the highest powers of all prime factors gives (2^3\times3^2=72). Step 3: For LCM, include both common and uncommon prime factors.
Correct answer: B. (20)
Explanation: Step 1: (40=2^3\times5) and (60=2^2\times3\times5). Step 2: The common factors with smaller powers are (2^2) and (5). Step 3: (2^2\times5=20), so the HCF is (20).
Correct answer: C. (120)
Explanation: Step 1: (40=2^3\times5) and (60=2^2\times3\times5). Step 2: Use the highest powers (2^3), (3), and (5). Step 3: (2^3\times3\times5=120), so the LCM is (120).
Correct answer: C. (15)
Explanation: Step 1: (45=3^2\times5) and (75=3\times5^2). Step 2: Take the smaller powers of the common prime factors (3) and (5). Step 3: (3\times5=15), so the HCF is (15).
Correct answer: B. (225)
Explanation: Step 1: (45=3^2\times5) and (75=3\times5^2). Step 2: Take the highest powers (3^2) and (5^2). Step 3: (3^2\times5^2=225), so the LCM is (225).
Correct answer: C. (16)
Explanation: Step 1: (32=2^5) and (48=2^4\times3). Step 2: The only common prime factor is (2), and the smaller power is (2^4). Step 3: (2^4=16), so the HCF is (16).
Correct answer: B. (96)
Explanation: Step 1: (32=2^5) and (48=2^4\times3). Step 2: Take the highest powers (2^5) and (3). Step 3: (2^5\times3=96), so the LCM is (96).
Correct answer: C. (9)
Explanation: Step 1: (27=3^3) and (36=2^2\times3^2). Step 2: The common prime factor is (3), and the smaller power is (3^2). Step 3: (3^2=9), so the HCF is (9).
Correct answer: C. (108)
Explanation: Step 1: (27=3^3) and (36=2^2\times3^2). Step 2: For LCM, take the highest powers (2^2) and (3^3). Step 3: (2^2\times3^3=108), so the LCM is (108).
Correct answer: C. (18)
Explanation: Step 1: (72=2^3\times3^2) and (90=2\times3^2\times5). Step 2: The smaller powers of common factors are (2) and (3^2). Step 3: (2\times9=18), so the HCF is (18).
Correct answer: C. (360)
Explanation: Step 1: (72=2^3\times3^2) and (90=2\times3^2\times5). Step 2: The highest powers are (2^3), (3^2), and (5). Step 3: (8\times9\times5=360), so the LCM is (360).
Correct answer: C. (6)
Explanation: Step 1: (12=2^2\times3), (18=2\times3^2), and (30=2\times3\times5). Step 2: The common prime factors in all three numbers are (2) and (3). Step 3: (2\times3=6), so the HCF is (6).
Correct answer: D. (180)
Explanation: Step 1: (12=2^2\times3), (18=2\times3^2), and (30=2\times3\times5). Step 2: Use the highest powers (2^2), (3^2), and (5). Step 3: (4\times9\times5=180), so the LCM is (180).
Correct answer: B. (8)
Explanation: Step 1: (16=2^4), (24=2^3\times3), and (40=2^3\times5). Step 2: The common prime factor is (2), and the smallest power is (2^3). Step 3: (2^3=8), so the HCF is (8).
Correct answer: C. (240)
Explanation: Step 1: (16=2^4), (24=2^3\times3), and (40=2^3\times5). Step 2: Take the highest powers (2^4), (3), and (5). Step 3: (16\times3\times5=240), so the LCM is (240).
Correct answer: B. (30)
Explanation: Step 1: The common prime factors are (2), (3), and (5). Step 2: Their smaller powers are (2), (3), and (5). Step 3: (2\times3\times5=30), so the HCF is (30).
Correct answer: C. (180)
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^2), (3^2), and (5). Step 3: (4\times9\times5=180), so the LCM is (180).
Correct answer: A. (12)
Explanation: Step 1: The common prime factors are (2) and (3). Step 2: The smaller powers are (2^2) and (3). Step 3: (2^2\times3=12), so the HCF is (12).
Correct answer: D. (72)
Explanation: Step 1: For LCM, take the highest powers. Step 2: Here the highest powers are (2^3) and (3^2). Step 3: (2^3\times3^2=72), so the LCM is (72).
Correct answer: C. (1620)
Explanation: Step 1: For two numbers, product equals HCF multiplied by LCM. Step 2: (9\times180=1620). Step 3: In such questions, first apply this relation and then multiply carefully.
Correct answer: B. (1728)
Explanation: Step 1: Product of two numbers (=) HCF (\times) LCM. Step 2: (12\times144=1728). Step 3: While multiplying, avoid missing any digit.
Correct answer: D. (864)
Explanation: Step 1: For two numbers, HCF (\times) LCM equals the product of the numbers. Step 2: (24\times36=864). Step 3: In such questions, you need not always find HCF and LCM separately.
Correct answer: B. (60)
Explanation: Step 1: Use the relation product (=) HCF (\times) LCM. Step 2: LCM (=\frac{720}{12}=60). Step 3: You can check the answer by multiplying (12) and (60).
Correct answer: B. (6)
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: HCF (=\frac{540}{90}=6). Step 3: In relation-based questions, place the given values carefully.
