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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. (11)
Explanation: Step 1: (22=2\times11) and (33=3\times11). Step 2: The common prime factor is (11). Step 3: Therefore, the HCF is (11).
Correct answer: B. (66)
Explanation: Step 1: (22=2\times11) and (33=3\times11). Step 2: For LCM, include (2), (3), and (11). Step 3: (2\times3\times11=66), so the answer is (66).
Correct answer: B. (6)
Explanation: Step 1: (30=2\times3\times5) and (42=2\times3\times7). Step 2: The common prime factors are (2) and (3). Step 3: (2\times3=6), so the HCF is (6).
Correct answer: C. (210)
Explanation: Step 1: (30=2\times3\times5) and (42=2\times3\times7). Step 2: Take (2), (3), (5), and (7) with highest powers. Step 3: (2\times3\times5\times7=210), so the LCM is (210).
Correct answer: C. (28)
Explanation: Step 1: (56=2^3\times7) and (84=2^2\times3\times7). Step 2: The smaller powers of common factors are (2^2) and (7). Step 3: (4\times7=28), so the HCF is (28).
Correct answer: B. (168)
Explanation: Step 1: (56=2^3\times7) and (84=2^2\times3\times7). Step 2: The highest powers are (2^3), (3), and (7). Step 3: (8\times3\times7=168), so the answer is (168).
Correct answer: C. (27)
Explanation: Step 1: (81=3^4) and (108=2^2\times3^3). Step 2: The common prime factor is (3), and the smaller power is (3^3). Step 3: (3^3=27), so the HCF is (27).
Correct answer: B. (324)
Explanation: Step 1: (81=3^4) and (108=2^2\times3^3). Step 2: For LCM, take (2^2) and (3^4). Step 3: (4\times81=324), so the LCM is (324).
Correct answer: C. (45)
Explanation: Step 1: (45=3^2\times5) and (90=2\times3^2\times5). Step 2: All prime factors of (45) are present in (90). Step 3: Therefore, the HCF is (45).
Correct answer: B. (90)
Explanation: Step 1: (45=3^2\times5) and (90=2\times3^2\times5). Step 2: The number (90) already contains all factors of (45). Step 3: So the LCM is (90).
Correct answer: B. (12)
Explanation: Step 1: (24=2^3\times3), (36=2^2\times3^2), and (48=2^4\times3). Step 2: The common smaller powers are (2^2) and (3). Step 3: (4\times3=12), so the HCF is (12).
Correct answer: C. (144)
Explanation: Step 1: (24=2^3\times3), (36=2^2\times3^2), and (48=2^4\times3). Step 2: The highest powers are (2^4) and (3^2). Step 3: (16\times9=144), so the LCM is (144).
Correct answer: B. (10)
Explanation: Step 1: (20=2^2\times5), (50=2\times5^2), and (70=2\times5\times7). Step 2: The common factors in all three are (2) and (5). Step 3: (2\times5=10), so the HCF is (10).
Correct answer: C. (700)
Explanation: Step 1: (20=2^2\times5), (50=2\times5^2), and (70=2\times5\times7). Step 2: The highest powers are (2^2), (5^2), and (7). Step 3: (4\times25\times7=700), so the answer is (700).
Correct answer: C. (21)
Explanation: Step 1: (42=2\times3\times7), (63=3^2\times7), and (105=3\times5\times7). Step 2: The common prime factors are (3) and (7). Step 3: (3\times7=21), so the answer is (21).
Correct answer: C. (630)
Explanation: Step 1: (42=2\times3\times7), (63=3^2\times7), and (105=3\times5\times7). Step 2: The highest powers are (2), (3^2), (5), and (7). Step 3: (2\times9\times5\times7=630), so the LCM is (630).
Correct answer: B. (60)
Explanation: Step 1: The common prime factors are (2), (3), and (5). Step 2: The smaller powers are (2^2), (3), and (5). Step 3: (4\times3\times5=60), so the HCF is (60).
Correct answer: C. (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. (24)
Explanation: Step 1: The common factors are (2) and (3). Step 2: The smaller powers are (2^3) and (3). Step 3: (2^3\times3=24), so the HCF is (24).
Correct answer: C. (288)
Explanation: Step 1: For LCM, choose the highest powers. Step 2: Here the highest powers are (2^5) and (3^2). Step 3: (32\times9=288), so the LCM is (288).
Correct answer: C. (1078)
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (7\times154=1078). Step 3: In such questions, knowing the two numbers separately is not necessary.
Correct answer: C. (3840)
Explanation: Step 1: The product of two numbers equals the product of their HCF and LCM. Step 2: (16\times240=3840). Step 3: Writing the relation first reduces calculation mistakes.
Correct answer: B. (84)
Explanation: Step 1: Use the relation product (=) HCF (\times) LCM. Step 2: LCM (=\frac{2016}{24}=84). Step 3: While dividing, simplify the division carefully.
Correct answer: B. (9)
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: HCF (=\frac{2700}{300}=9). Step 3: You can check the answer using (9\times300=2700).
Correct answer: C. (5184)
Explanation: Step 1: For two numbers, HCF (\times) LCM equals the product of the two numbers. Step 2: (54\times96=5184). Step 3: This shortcut saves time in exams.
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