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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: A. (1)
Explanation: Step 1: (14=2\times7) and (25=5^2). Step 2: They have no common prime factor. Step 3: Therefore, the HCF is (1), and such numbers are co-prime.
Correct answer: C. (350)
Explanation: Step 1: (14=2\times7) and (25=5^2). Step 2: Since they have no common prime factor, include all factors for LCM. Step 3: (2\times7\times25=350), so the LCM is (350).
Correct answer: D. (120)
Explanation: Step 1: (8=2^3) and (15=3\times5). Step 2: They have no common prime factor. Step 3: The LCM of co-prime numbers is their product, so (8\times15=120).
Correct answer: C. (7)
Explanation: Step 1: (21=3\times7) and (35=5\times7). Step 2: The common prime factor is (7). Step 3: Therefore, the HCF is (7).
Correct answer: C. (105)
Explanation: Step 1: (21=3\times7) and (35=5\times7). Step 2: For LCM, take (3), (5), and (7). Step 3: (3\times5\times7=105), so the LCM is (105).
Correct answer: B. (12)
Explanation: Step 1: For the maximum number of equal packets, find the HCF. Step 2: (24=2^3\times3) and (36=2^2\times3^2), so HCF (=2^2\times3=12). Step 3: When maximum equal groups are asked, use HCF.
Correct answer: C. (15)
Explanation: Step 1: For maximum equal boxes, find the HCF of (30), (45), and (60). Step 2: (30=2\times3\times5), (45=3^2\times5), (60=2^2\times3\times5). Step 3: The common part is (3\times5=15), so (15) boxes can be made.
Correct answer: C. (36)
Explanation: Step 1: The time when both bells ring together is the LCM of the two times. Step 2: (12=2^2\times3) and (18=2\times3^2), so LCM (=2^2\times3^2=36). Step 3: For repeated time events, use LCM.
Correct answer: B. (60)
Explanation: Step 1: The next common flashing time is the LCM of (10), (15), and (20). Step 2: (10=2\times5), (15=3\times5), (20=2^2\times5). Step 3: (2^2\times3\times5=60), so the answer is (60) seconds.
Correct answer: C. (24) cm
Explanation: Step 1: For equal pieces of maximum length, find the HCF. Step 2: (48=2^4\times3) and (72=2^3\times3^2), so HCF (=2^3\times3=24). Step 3: For cutting or dividing into maximum equal parts, use HCF.
Correct answer: B. (4) m
Explanation: Step 1: For maximum equal length, find the HCF of (20) and (28). Step 2: (20=2^2\times5) and (28=2^2\times7). Step 3: The common part is (2^2=4), so each piece will be (4) m long.
Correct answer: C. (27)
Explanation: Step 1: (54=2\times3^3) and (81=3^4). 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: A. (162)
Explanation: Step 1: (54=2\times3^3) and (81=3^4). Step 2: Take the highest powers (2) and (3^4). Step 3: (2\times81=162), so the LCM is (162).
Correct answer: C. (21)
Explanation: Step 1: (63=3^2\times7) and (84=2^2\times3\times7). Step 2: The common factors are (3) and (7). Step 3: (3\times7=21), so the HCF is (21).
Correct answer: C. (252)
Explanation: Step 1: (63=3^2\times7) and (84=2^2\times3\times7). Step 2: The highest powers are (2^2), (3^2), and (7). Step 3: (4\times9\times7=252), so the LCM is (252).
Correct answer: B. (24)
Explanation: Step 1: The common prime 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. (480)
Explanation: Step 1: The LCM contains the highest powers of all prime factors. Step 2: The highest powers are (2^5), (3), and (5). Step 3: (32\times3\times5=480), so the answer is (480).
Correct answer: A. (5)
Explanation: Step 1: (25=5^2), (40=2^3\times5), and (50=2\times5^2). Step 2: The common prime factor in all three numbers is (5). Step 3: The smallest power is (5), so the HCF is (5).
Correct answer: B. (200)
Explanation: Step 1: (25=5^2), (40=2^3\times5), and (50=2\times5^2). Step 2: The highest powers are (2^3) and (5^2). Step 3: (8\times25=200), so the LCM is (200).
Correct answer: D. (44)
Explanation: Step 1: (88=2^3\times11) and (132=2^2\times3\times11). Step 2: The smaller powers of common factors are (2^2) and (11). Step 3: (4\times11=44), so the HCF is (44).
Correct answer: B. (264)
Explanation: Step 1: (88=2^3\times11) and (132=2^2\times3\times11). Step 2: The highest powers are (2^3), (3), and (11). Step 3: (8\times3\times11=264), so the LCM is (264).
Correct answer: C. (84)
Explanation: Step 1: First simplify the given prime powers. Step 2: Since (2^2=4), the number is (4\times3\times7=84). Step 3: Reading prime factorisation correctly helps in HCF and LCM questions.
Correct answer: B. (200)
Explanation: Step 1: (2^3=8) and (5^2=25). Step 2: Multiplying them gives (8\times25=200). Step 3: Always simplify powers first and then multiply.
Correct answer: A. They are co-prime
Explanation: Step 1: If the HCF of two numbers is (1), they have no common factor except (1). Step 2: Such numbers are called co-prime. Step 3: Co-prime numbers need not both be prime numbers.
Correct answer: A. Smaller power of common factors
Explanation: Step 1: HCF is made only from common prime factors. Step 2: Among common factors, the smaller power is chosen because it divides both numbers. Step 3: Remember the rule: common factors with smaller powers for HCF.
