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SubjectsMathematics
अभाज्य गुणनखंडन
In Class 10 Mathematics, under the Real Numbers chapter, Prime Factorisation teaches students to express a composite number as a product of prime numbers. Students practise identifying prime factors and writing factorisations using multiplication and exponents. This foundational skill supports the Fundamental Theorem of Arithmetic and later work with HCF and LCM.
Correct answer: A. (24)
Explanation: Step 1: For prime factorisation (p^a q^b r^c), total factors are ((a+1)(b+1)(c+1)). Step 2: Here the powers are (3,2,1), so factors (=(4)(3)(2)=24). Step 3: In exams, always add (1) to each exponent before multiplying.
Correct answer: A. (3)
Explanation: Step 1: In a perfect square, every prime factor must have an even exponent. Step 2: (2^4) and (5^2) are already even powers, but (3^3) is odd. Dividing by (3) leaves (3^2). Step 3: Fix only the prime factor with an odd exponent.
Correct answer: A. (2\times3\times7^2)
Explanation: Step 1: In a perfect cube, every prime exponent must be a multiple of (3). Step 2: (2^5) needs one more (2), (3^2) needs one more (3), and (7^1) needs (7^2). Step 3: Complete each exponent to the next multiple of (3).
Correct answer: A. (2^2\times3^2)
Explanation: Step 1: HCF contains only common prime factors. Step 2: The smaller power of (2) is (2), and the smaller power of (3) is (2). So HCF (=2^2\times3^2). Step 3: For HCF, choose the smaller exponent of common primes.
Correct answer: A. (2^3\times3^2\times5^2)
Explanation: Step 1: LCM includes every prime factor with its highest exponent. Step 2: Highest powers are (2^3), (3^2), and (5^2). Step 3: For LCM, do not miss a prime factor that appears in either number.
Correct answer: A. (2^2\times3^2\times5)
Explanation: Step 1: LCM uses higher exponents and HCF uses lower exponents. Step 2: In the ratio, subtract exponents: (2^{4-2}\times3^{3-1}\times5^{2-1}=2^2\times3^2\times5). Step 3: Use exponent difference instead of calculating both full numbers.
Correct answer: A. (588)
Explanation: Step 1: Multiply the prime factors according to their powers. Step 2: (2^2=4) and (7^2=49), so (4\times3\times49=588). Step 3: First evaluate powers, then multiply carefully.
Correct answer: A. (3)
Explanation: Step 1: Total factors are ((a+1)(2+1)(1+1)). Step 2: ((a+1)\times3\times2=24), so (a+1=4) and (a=3). Step 3: For an unknown exponent, apply the factor-count rule directly.
Correct answer: A. (36)
Explanation: Step 1: A factor divisible by (2) must contain (2^1) at least. Step 2: Powers of (2) can be (1,2,3), giving (3) choices; powers of (3) give (3) choices; (5) and (7) give (2) choices each. Total (=3\times3\times2\times2=36). Step 3: For conditional factors, adjust only the restricted prime exponent.
Correct answer: A. (6)
Explanation: Step 1: An odd factor must not contain the prime (2). Step 2: Power of (2) is only (0); power of (3) has (3) choices and power of (5) has (2) choices. Total (=3\times2=6). Step 3: When counting odd factors, ignore the power choices of (2) except zero.
Correct answer: A. (1)
Explanation: Step 1: Coprime numbers do not have any common factor greater than (1). Step 2: Therefore, their HCF is always (1), even if their product is large. Step 3: When you see coprime, think about common factors first.
Correct answer: A. Every integer greater than (1) can be written as a product of primes in one basic way
Explanation: Step 1: The fundamental theorem of arithmetic says every integer greater than (1) has a unique prime factorisation except for order. Step 2: Option A states this correctly. Step 3: Unique means the same prime factors appear, only their order may change.
Correct answer: A. (2^3\times3^2\times5)
Explanation: Step 1: In square root, each prime exponent becomes half. Step 2: (2^6) becomes (2^3), (3^4) becomes (3^2), and (5^2) becomes (5). Step 3: This direct method works when all exponents are even.
Correct answer: A. (2^3\times3^2\times7)
Explanation: Step 1: For cube root, divide each prime exponent by (3). Step 2: (9\div3=3), (6\div3=2), and (3\div3=1), so the cube root is (2^3\times3^2\times7). Step 3: In a perfect cube, all exponents are multiples of (3).
Correct answer: A. (3\times5)
Explanation: Step 1: A perfect square needs all exponents to be even. Step 2: (2^2) is already fine, (3^3) needs one (3), and (5^1) needs one (5). Step 3: Make only the odd exponents even.
Correct answer: A. (2\times3^2\times11^2)
Explanation: Step 1: Division reduces exponents, and a perfect cube needs remaining exponents as multiples of (3). Step 2: Divide (2^4) by (2), (3^5) by (3^2), and (11^2) by (11^2). Step 3: In division questions, reduce each exponent to the nearest lower multiple of (3).
Correct answer: A. (2^7\times3^4\times5)
Explanation: Step 1: Product of two numbers equals HCF multiplied by LCM. Step 2: Multiply (2^2\times3) with (2^5\times3^3\times5); add exponents to get (2^7\times3^4\times5). Step 3: When multiplying same bases, add exponents.
Correct answer: A. (2^2\times3^3\times7)
Explanation: Step 1: Write (756) as (4\times189). Step 2: (4=2^2) and (189=27\times7=3^3\times7), so (756=2^2\times3^3\times7). Step 3: Break a large number into convenient factors first.
Correct answer: A. (18)
Explanation: Step 1: Since (10=2\times5), the factor must contain at least one (2) and one (5). Step 2: Powers of (2): (1,2,3) give (3) choices; powers of (3): (0,1,2) give (3) choices; powers of (5): (1,2) give (2) choices. Total (=18). Step 3: For divisibility, check the required minimum prime powers.
Correct answer: A. (22)
Explanation: Step 1: Total factors are ((5+1)(4+1)(1+1)=60). Step 2: Factors divisible by (6=2\times3) must have power of (2) at least (1) and power of (3) at least (1), so (5\times4\times2=40). Step 3: Not divisible by (6) means total minus divisible factors, (60-40=20).
Correct answer: A. (4)
Explanation: Step 1: Total factors are ((r+1)(2+1)). Step 2: ((r+1)\times3=15), so (r+1=5) and (r=4). Step 3: Write the factor-count formula first, then solve the simple equation.
Correct answer: A. (2\times5)
Explanation: Step 1: A perfect square needs every exponent even. Step 2: (2^3) and (5^1) have odd powers, so multiply by (2\times5). Step 3: Do not multiply the prime whose exponent is already even.
Correct answer: A. (1:(2\times3^2\times11))
Explanation: Step 1: HCF is (2^1\times3^1\times11^1), and LCM is (2^2\times3^3\times11^2). Step 2: Dividing both sides by the HCF gives (1:2^1\times3^2\times11^1). Step 3: Simplifying the ratio is important in such questions.
Correct answer: A. (16)
Explanation: Step 1: An even factor must contain at least one (2). Step 2: Power of (2) has (2) choices, (1) or (2); each of (3,5,7) has (2) choices. Total (=16). Step 3: For even factors, exclude the case (2^0).
Correct answer: A. (2\times3^2\times5\times11)
Explanation: Step 1: Write (990=99\times10). Step 2: (99=3^2\times11) and (10=2\times5), so (990=2\times3^2\times5\times11). Step 3: Split the number into easy parts for faster factorisation.
