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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. (32)
Explanation: Step 1: Since (15=3\times5), the factor must contain both (3) and (5). Step 2: Power choices are (2:4) choices, (3:2) choices, (5:2) choices, and (7:2) choices. Total (=4\times2\times2\times2=32). Step 3: Start restricted prime powers from the minimum required value.
Correct answer: A. (a=2,\ b=5)
Explanation: Step 1: Total factors are ((a+1)(b+1)). Step 2: For (a=2,\ b=5), we get ((3)(6)=18). The other options do not give (18). Step 3: In option-based questions, substitute values into the rule quickly.
Correct answer: A. (2^2\times3\times5^2)
Explanation: Step 1: HCF uses lower exponents and LCM uses higher exponents. Step 2: Exponent differences give (2^{4-2}\times3^{2-1}\times5^{3-1}=2^2\times3\times5^2). Step 3: For how many times, divide LCM by HCF.
Correct answer: D. None of the values is possible
Explanation: Step 1: A perfect square requires all prime exponents to be even. Step 2: The exponent of (2) is already (3), which is odd, so changing only (x) cannot make the number a perfect square. Step 3: Check the whole prime factorisation, not only the unknown exponent.
Correct answer: A. (2)
Explanation: Step 1: Total factors are ((x+1)(2+1)(4+1)). Step 2: ((x+1)\times3\times5=45), so (x+1=3) and (x=2). Step 3: Divide the given factor count by the known parts first.
Correct answer: A. (2^2\times3^2\times5\times7)
Explanation: Step 1: Write (1260=126\times10). Step 2: (126=2\times3^2\times7) and (10=2\times5), so (1260=2^2\times3^2\times5\times7). Step 3: Combine repeated prime factors into powers at the end.
Correct answer: A. (4)
Explanation: Step 1: A square factor must have even exponents for every prime. Step 2: For (2), choices are (0,2); for (3), choices are (0,2); for (5), only (0). Total (=2\times2\times1=4). Step 3: Count only even exponent choices for square factors.
Correct answer: A. (2)
Explanation: Step 1: A cube factor must have exponents that are multiples of (3). Step 2: For (2), only (0); for (3), (0) or (3); for (5), only (0). Total (=2). Step 3: Remember that (0) is also a multiple of (3).
Correct answer: A. (2^5\times3^3\times5^3)
Explanation: Step 1: When multiplying powers with the same prime base, add exponents. Step 2: (2^{3+2}\times3^{1+2}\times5^{1+2}=2^5\times3^3\times5^3). Step 3: Add exponents for multiplication with the same base.
Correct answer: A. (2^4\times3^3\times5)
Explanation: Step 1: When dividing powers with the same base, subtract exponents. Step 2: (2^{6-2}\times3^{4-1}\times5^{2-1}=2^4\times3^3\times5). Step 3: In division, subtract the smaller exponent from the larger one.
Correct answer: A. (48)
Explanation: Step 1: A factor divisible by (5) must have power of (5) at least (1). Step 2: Choices are (4) for (2), (3) for (3), and (4) for (5). Total (=4\times3\times4=48). Step 3: Do not include zero power for the required prime.
Correct answer: A. (72)
Explanation: Step 1: For total factors, add (1) to each exponent. Step 2: ((2+1)(3+1)(1+1)(2+1)=3\times4\times2\times3=72). Step 3: The same rule works even when many prime factors are present.
Correct answer: A. (2^2\times3\times5^2)
Explanation: Step 1: After division, remaining exponents must be multiples of (3). Step 2: Remove (2^2), remove (3) from (3^4), and remove (5^2) from (5^5). Step 3: Reduce each exponent to the nearest lower multiple of (3).
Correct answer: A. (7)
Explanation: Step 1: Total factors are ((3+1)(2+1)(x+1)). Step 2: (4\times3\times(x+1)=96), so (x+1=8) and (x=7). Step 3: Multiply known parts first, then solve for the unknown exponent.
Correct answer: A. (2^4\times3^3\times5)
Explanation: Step 1: Product of two numbers (=) HCF (\times) LCM. Step 2: (12=2^2\times3) and (180=2^2\times3^2\times5), so the product is (2^4\times3^3\times5). Step 3: Convert given numbers to prime form before multiplying.
Correct answer: A. (30)
Explanation: Step 1: Since (25=5^2), the factor must contain at least (5^2). Step 2: Choices for (2): (5), for (3): (3), for (5): (2) or (3), giving (2) choices. Total (=5\times3\times2=30). Step 3: Treat (25) as (5^2) before counting.
Correct answer: A. (48)
Explanation: Step 1: (12=2^2\times3), so the factor needs power of (2) at least (2) and power of (3) at least (1). Step 2: Choices are (4) for (2), (3) for (3), and (3) for (5). Total (=4\times3\times3=36). Step 3: First write the divisor in prime form, then set exponent limits.
Correct answer: A. ((2,1,1))
Explanation: Step 1: In division, exponents of the same base are subtracted. Step 2: (2^{2-a}=2^0) gives (a=2), (3^{3-b}=3^2) gives (b=1), and (5^{2-c}=5^1) gives (c=1). Step 3: If a prime is not visible in the result, treat its exponent as (0).
Correct answer: A. (2^4\times3^2\times5^3)
Explanation: Step 1: Add (1) to each exponent and multiply to get total factors. Step 2: In option A, ((4+1)(2+1)(3+1)=5\times3\times4=60). Step 3: In option-based questions, test the exponents of each option.
Correct answer: A. (12)
Explanation: Step 1: Since (9=3^2), power of (3) must be at least (2). Not divisible by (5) means power of (5) must be (0). Step 2: Choices are (4) for (2), (3) for (3), and (1) for (5). Total (=12). Step 3: Convert each condition into exponent restrictions.
Correct answer: A. (32)
Explanation: Step 1: If (d=2^a\times3^b\times7^c), then (d^2=2^{2a}\times3^{2b}\times7^{2c}). Step 2: Conditions are (2a\le6), (2b\le2), (2c\le2), so choices are (4,2,2). Total (=16). Step 3: For square divisibility, double the exponents and compare.
Correct answer: A. (8)
Explanation: Step 1: Let (d=2^a\times3^b\times5^c), so (d^3=2^{3a}\times3^{3b}\times5^{3c}). Step 2: (3a\le5), (3b\le4), and (3c\le3), giving (2) choices each. Total (=8). Step 3: For cube divisibility, triple the exponents and compare.
Correct answer: A. (3)
Explanation: Step 1: To be divisible by neither (2) nor (3), powers of (2) and (3) must both be (0). Step 2: Power of (5) can be (0,1,2), giving (3) factors. Step 3: For neither-nor conditions, set both restricted prime powers to zero.
Correct answer: A. (12)
Explanation: Step 1: HCF uses the smaller exponents. Step 2: HCF (=2^3\times3^2). Its number of factors is ((3+1)(2+1)=12). Step 3: First find the HCF, then count its factors.
Correct answer: A. (45)
Explanation: Step 1: LCM takes the highest exponent of every prime factor. Step 2: LCM (=2^4\times3^2\times5^3). Total factors (=(4+1)(2+1)(3+1)=5\times3\times4=60). Step 3: After finding the LCM, apply the factor-count rule.
