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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. (2^4\times3^2)
Explanation: Step 1: HCF includes only common primes (2) and (3). Step 2: The smaller powers are (2^4) and (3^2), so (M=2^4\times3^2). Step 3: Do not include (5) and (7), as they appear in only one number.
Correct answer: C. (6)
Explanation: Step 1: Compare the powers of (2) separately. Step 2: (40=2^3\times5), (64=2^6), and (96=2^5\times3), so the power of (2) in the LCM is (6). Step 3: You can compare powers before calculating the full LCM.
Correct answer: B. (2835)
Explanation: Step 1: The smallest common divisible number is the LCM. Step 2: (27=3^3), (45=3^2\times5), and (63=3^2\times7), so LCM (=3^3\times5\times7=945). Step 3: Calculate before choosing, because larger options can mislead.
Correct answer: A. Such two whole numbers are not possible
Explanation: Step 1: The HCF must divide the LCM. Step 2: (180) is not exactly divisible by (9), so such whole numbers are not possible. Step 3: Check this necessary condition before searching for pairs.
Correct answer: B. (42)
Explanation: Step 1: When the HCF is (12), (m) and (n) are coprime. Step 2: LCM (=12mn=504), so (mn=42). Step 3: This form makes large-number questions easier.
Correct answer: B. LCM
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: So product divided by HCF gives the LCM. Step 3: Learn to rearrange the relation, not just memorize it.
Correct answer: A. (15:1)
Explanation: Step 1: HCF is (2^2\times3^2=36). Step 2: LCM is (2^2\times3^3\times5=540), so the ratio is (540:36=15:1). Step 3: Always reduce the ratio to its simplest form.
Correct answer: C. (66)
Explanation: Step 1: The maximum number of rows is found by HCF. Step 2: (132=2^2\times3\times11) and (198=2\times3^2\times11), so HCF (=2\times3\times11=66). Step 3: For maximum equal arrangement, identify HCF.
Correct answer: B. Such numbers are not possible
Explanation: Step 1: The HCF must be an exact divisor of the LCM. Step 2: (1260) is not exactly divisible by (45), so such two whole numbers are not possible. Step 3: This quick check saves long calculations.
Correct answer: A. (2^5\times3^4\times5\times7)
Explanation: Step 1: A common multiple must be a multiple of the LCM. Step 2: The LCM contains (2^5), (3^4), (5), and (7). Step 3: For a multiple, every required prime power must be present.
Correct answer: A. (42) and (231)
Explanation: Step 1: (42=21\times2) and (231=21\times11). Step 2: Since (2) and (11) are coprime, HCF is (21) and LCM is (21\times2\times11=462). Step 3: Factor out the HCF and check whether the remaining numbers are coprime.
Correct answer: A. (a=3) can be true
Explanation: Step 1: The smaller power of (2) in the HCF must be (3). Step 2: The second number has power (5), so (a=3) makes the smaller power (3). Step 3: In power questions, compare smaller and larger values carefully.
Correct answer: B. (1)
Explanation: Step 1: Compare the powers of (3). Step 2: (72=2^3\times3^2), (90=2\times3^2\times5), and (150=2\times3\times5^2), so the smallest power is (1). Step 3: HCF uses the smallest power.
Correct answer: C. (4)
Explanation: Step 1: The smallest such number is the LCM of the two given numbers. Step 2: The higher power of (3) is (4), so the smallest number contains (3^4). Step 3: For divisibility, check powers in the LCM.
Correct answer: A. (21)
Explanation: Step 1: (a=2^2\times5=20). Step 2: Since (ab=420), (b=\frac{420}{20}=21), and (20) and (21) are coprime. Step 3: The coprime condition helps verify the final answer.
Correct answer: C. (4)
Explanation: Step 1: Product of two numbers equals HCF times LCM. Step 2: The powers of (3) are (1) and (3), so the total power is (4). Step 3: When multiplying powers with the same base, add the exponents.
Correct answer: A. (12)
Explanation: Step 1: Prime factorise: (48=2^4\times3), (72=2^3\times3^2), and (108=2^2\times3^3). Step 2: The common smallest powers are (2^2) and (3), so HCF is (12). Step 3: For three numbers, take the smallest power across all.
Correct answer: B. (810)
Explanation: Step 1: (54=2\times3^3), (81=3^4), and (135=3^3\times5). Step 2: Using highest powers, LCM (=2\times3^4\times5=810). Step 3: A prime appearing in only one number still appears in the LCM.
Correct answer: C. (168)
Explanation: Step 1: Product of two numbers equals HCF times LCM. Step 2: The other number is (\frac{28\times840}{140}=168). Step 3: Simplify the division to reduce mistakes.
Correct answer: A. (1200)
Explanation: Step 1: Product equals HCF (\times) LCM. Step 2: LCM (=\frac{24000}{20}=1200). Step 3: Check the answer using the given relation.
Correct answer: C. (350)
Explanation: Step 1: (75=3\times5^2) and (125=5^3). Step 2: HCF (=5^2=25) and LCM (=3\times5^3=375), so (L-H=350). Step 3: Find both values separately first.
Correct answer: A. (2^2\times3\times5)
Explanation: Step 1: The ratio means dividing LCM by HCF. Step 2: Subtract powers of the same bases: (2^{4-2}\times3^{2-1}\times5=2^2\times3\times5). Step 3: Remember the exponent rule for division.
Correct answer: A. (336)
Explanation: Step 1: When (r) and (s) are coprime, LCM is (6rs). Step 2: (7) and (8) are coprime, so LCM (=6\times7\times8=336). Step 3: Factor out the HCF and check the remaining numbers.
Correct answer: B. (1)
Explanation: Step 1: LCM contains the highest power of every prime present. Step 2: (11) appears only in the second number as (11^1), so its power in (L) is (1). Step 3: A prime appearing in only one number is still included in the LCM.
Correct answer: C. (540)
Explanation: Step 1: Remainder (0) means the number is exactly divisible by all three numbers. Step 2: (36=2^2\times3^2), (54=2\times3^3), and (90=2\times3^2\times5), so LCM (=2^2\times3^3\times5=540). Step 3: The smallest possible value is always the LCM.
