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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: (16=2^4) and (27=3^3). Step 2: They have no common prime factor. Step 3: So the HCF is (1), and these numbers are co-prime.
Correct answer: D. (432)
Explanation: Step 1: (16=2^4) and (27=3^3). Step 2: There is no common factor, so the LCM is their product. Step 3: (16\times27=432), so the answer is (432).
Correct answer: C. (180)
Explanation: Step 1: (9=3^2) and (20=2^2\times5). Step 2: They have no common prime factor. Step 3: The LCM of co-prime numbers is their product, so (9\times20=180).
Correct answer: B. (13)
Explanation: Step 1: (26=2\times13) and (39=3\times13). Step 2: The common prime factor is (13). Step 3: Therefore, the HCF is (13).
Correct answer: A. (78)
Explanation: Step 1: (26=2\times13) and (39=3\times13). Step 2: For LCM, take (2), (3), and (13). Step 3: (2\times3\times13=78), so the answer is (78).
Correct answer: C. (16)
Explanation: Step 1: For maximum equal groups, find the HCF. Step 2: (32=2^5) and (48=2^4\times3), so HCF (=2^4=16). Step 3: In equal distribution questions, the word maximum usually points to HCF.
Correct answer: B. (14)
Explanation: Step 1: For the maximum number of equal bags, find the HCF of (42) and (56). Step 2: (42=2\times3\times7) and (56=2^3\times7), so HCF (=2\times7=14). Step 3: In packing questions with equal and maximum, use HCF.
Correct answer: C. (60)
Explanation: Step 1: The next common ringing time is the LCM of the intervals. Step 2: (15=3\times5) and (20=2^2\times5), so LCM (=2^2\times3\times5=60). Step 3: For repeated time events, use LCM.
Correct answer: C. (48)
Explanation: Step 1: The next common glowing time is the LCM of (8), (12), and (16). Step 2: (8=2^3), (12=2^2\times3), and (16=2^4). Step 3: (2^4\times3=48), so the answer is (48) seconds.
Correct answer: B. (12)
Explanation: Step 1: For equal pieces of maximum length, find the HCF. Step 2: (60=2^2\times3\times5) and (84=2^2\times3\times7), so HCF (=2^2\times3=12). Step 3: In cutting questions, maximum equal length is found by HCF.
Correct answer: B. (18)
Explanation: Step 1: For maximum equal length, find the HCF of (54) and (90). Step 2: (54=2\times3^3) and (90=2\times3^2\times5), so HCF (=2\times3^2=18). Step 3: When lengths are cut equally, use HCF.
Correct answer: C. (25)
Explanation: Step 1: (75=3\times5^2) and (100=2^2\times5^2). Step 2: The common prime factor is (5), and the smaller power is (5^2). Step 3: (5^2=25), so the HCF is (25).
Correct answer: C. (300)
Explanation: Step 1: (75=3\times5^2) and (100=2^2\times5^2). Step 2: The highest powers are (2^2), (3), and (5^2). Step 3: (4\times3\times25=300), so the LCM is (300).
Correct answer: D. (49)
Explanation: Step 1: (98=2\times7^2) and (147=3\times7^2). Step 2: The common prime factor is (7), and the smaller power is (7^2). Step 3: (7^2=49), so the HCF is (49).
Correct answer: A. (294)
Explanation: Step 1: (98=2\times7^2) and (147=3\times7^2). Step 2: The LCM includes (2), (3), and (7^2). Step 3: (2\times3\times49=294), so the answer is (294).
Correct answer: B. (36)
Explanation: Step 1: The common prime factors are (2) and (3). Step 2: The smaller powers are (2^2) and (3^2). Step 3: (4\times9=36), so the HCF is (36).
Correct answer: C. (540)
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^2), (3^3), and (5). Step 3: (4\times27\times5=540), so the answer is (540).
Correct answer: C. (27)
Explanation: Step 1: (27=3^3), (54=2\times3^3), and (81=3^4). Step 2: The common prime factor is (3), and the smallest power is (3^3). Step 3: (3^3=27), so the HCF is (27).
Correct answer: C. (162)
Explanation: Step 1: (27=3^3), (54=2\times3^3), and (81=3^4). Step 2: The highest powers are (2) and (3^4). Step 3: (2\times81=162), so the LCM is (162).
Correct answer: B. (24)
Explanation: Step 1: (72=2^3\times3^2) and (120=2^3\times3\times5). Step 2: The smaller powers of common factors are (2^3) and (3). Step 3: (8\times3=24), so the HCF is (24).
Correct answer: B. (360)
Explanation: Step 1: (72=2^3\times3^2) and (120=2^3\times3\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. (360)
Explanation: Step 1: First evaluate the powers: (2^3=8) and (3^2=9). Step 2: Now (8\times9\times5=360). Step 3: While forming a number from factorisation, simplify powers first and then multiply.
Correct answer: C. (1200)
Explanation: Step 1: (2^4=16) and (5^2=25). Step 2: The number is (16\times3\times25=1200). Step 3: In long multiplication, pair easy numbers first, such as (16\times25=400).
Correct answer: B. Highest power of all prime factors
Explanation: Step 1: LCM is the smallest number divisible by all the given numbers. Step 2: Therefore, the highest powers of all prime factors are taken. Step 3: Remember the rule: all factors with highest powers for LCM.
Correct answer: B. ( \text{HCF}\times\text{LCM}=\text{product of the two numbers} )
Explanation: Step 1: For two numbers, a special relation is used. Step 2: The product of HCF and LCM equals the product of the two numbers. Step 3: Apply this relation directly only for two numbers.
