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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. (10)
Explanation: Step 1: (20=2^2\times5) and (30=2\times3\times5). Step 2: The common prime factors are (2) and (5) with smaller powers (2) and (5). Step 3: (2\times5=10), so the HCF is (10).
Correct answer: C. (60)
Explanation: Step 1: (20=2^2\times5) and (30=2\times3\times5). Step 2: For LCM, take the highest powers (2^2), (3), and (5). Step 3: (4\times3\times5=60), so the LCM is (60).
Correct answer: B. (14)
Explanation: Step 1: (28=2^2\times7) and (42=2\times3\times7). Step 2: The common factors are (2) and (7). Step 3: (2\times7=14), so the HCF is (14).
Correct answer: B. (84)
Explanation: Step 1: (28=2^2\times7) and (42=2\times3\times7). Step 2: The highest powers are (2^2), (3), and (7). Step 3: (4\times3\times7=84), so the LCM is (84).
Correct answer: C. (18)
Explanation: Step 1: (36=2^2\times3^2) and (54=2\times3^3). 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. (108)
Explanation: Step 1: (36=2^2\times3^2) and (54=2\times3^3). Step 2: For LCM, take the highest powers (2^2) and (3^3). Step 3: (4\times27=108), so the answer is (108).
Correct answer: B. (22)
Explanation: Step 1: (44=2^2\times11) and (66=2\times3\times11). Step 2: The common prime factors are (2) and (11). Step 3: (2\times11=22), so the HCF is (22).
Correct answer: C. (132)
Explanation: Step 1: (44=2^2\times11) and (66=2\times3\times11). Step 2: Take the highest powers (2^2), (3), and (11). Step 3: (4\times3\times11=132), so the LCM is (132).
Correct answer: B. (7)
Explanation: Step 1: (49=7^2) and (70=2\times5\times7). Step 2: The only common prime factor is (7). Step 3: Therefore, the HCF is (7).
Correct answer: C. (490)
Explanation: Step 1: (49=7^2) and (70=2\times5\times7). Step 2: For LCM, take (2), (5), and the higher power (7^2). Step 3: (2\times5\times49=490), so the answer is (490).
Correct answer: B. (16)
Explanation: Step 1: (64=2^6) and (80=2^4\times5). Step 2: The common prime factor is (2), and the smaller power is (2^4). Step 3: (2^4=16), so the HCF is (16).
Correct answer: C. (320)
Explanation: Step 1: (64=2^6) and (80=2^4\times5). Step 2: The highest powers are (2^6) and (5). Step 3: (64\times5=320), so the LCM is (320).
Correct answer: B. (5)
Explanation: Step 1: (15=3\times5), (25=5^2), and (35=5\times7). Step 2: The common prime factor in all three numbers is (5). Step 3: Therefore, the HCF is (5).
Correct answer: C. (525)
Explanation: Step 1: (15=3\times5), (25=5^2), and (35=5\times7). Step 2: The highest powers are (3), (5^2), and (7). Step 3: (3\times25\times7=525), so the LCM is (525).
Correct answer: C. (9)
Explanation: Step 1: (18=2\times3^2), (27=3^3), and (45=3^2\times5). Step 2: The common prime factor is (3), and the smallest power is (3^2). Step 3: (3^2=9), so the answer is (9).
Correct answer: C. (270)
Explanation: Step 1: (18=2\times3^2), (27=3^3), and (45=3^2\times5). Step 2: The highest powers are (2), (3^3), and (5). Step 3: (2\times27\times5=270), so the LCM is (270).
Correct answer: B. (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. (360)
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^3), (3^2), and (5). Step 3: (8\times9\times5=360), so the answer is (360).
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: C. (336)
Explanation: Step 1: For LCM, choose the highest powers. Step 2: The highest powers are (2^4), (3), and (7). Step 3: (16\times3\times7=336), so the LCM is (336).
Correct answer: C. (768)
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (8\times96=768). Step 3: In such questions, apply the relation first and then multiply carefully.
Correct answer: C. (3150)
Explanation: Step 1: To find the product, multiply the HCF and LCM. Step 2: (15\times210=3150). Step 3: Use this relation directly for two numbers.
Correct answer: B. (72)
Explanation: Step 1: Use the relation product (=) HCF (\times) LCM. Step 2: LCM (=\frac{1296}{18}=72). Step 3: After division, you can check by multiplying (18\times72).
Correct answer: B. (6)
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: HCF (=\frac{1680}{280}=6). Step 3: It is important to place the given values correctly.
Correct answer: D. (2304)
Explanation: Step 1: For two numbers, HCF (\times) LCM equals the product of the two numbers. Step 2: (32\times72=2304). Step 3: In such questions, finding HCF and LCM separately is not necessary.
