Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है
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. Perfect square
Explanation: Step 1: In a perfect square, all prime exponents are even. Step 2: Here every exponent is (2), so (N) is a perfect square. Step 3: To identify a perfect square, check whether exponents are even.
Correct answer: B. Perfect cube
Explanation: Step 1: In a perfect cube, every prime exponent is a multiple of (3). Step 2: Both (6) and (3) are multiples of (3), so (N) is a perfect cube. Step 3: For a perfect cube, check exponents using (3).
Correct answer: C. 135
Explanation: Step 1: An odd factor must not contain (2). Step 2: After removing (2^2), we get (3^3 \times 5=27 \times 5=135). Step 3: To get the greatest odd factor, remove all powers of (2).
Correct answer: B. (2 \times 3^2 \times 5 \times 7)
Explanation: Step 1: The given number has no (2), so it is odd. Step 2: To make the smallest even multiple, multiplying by one (2) is enough. Step 3: When the smallest multiple is asked, do not increase exponents unnecessarily.
Correct answer: A. Because all exponents are not even
Explanation: Step 1: The square root of a number is an integer only when all prime exponents are even. Step 2: In (2^5 \times 3), the exponents are (5) and (1), both odd. Step 3: For square-root questions, check evenness of exponents first.
Correct answer: A. (2^2 \times 3)
Explanation: Step 1: In a cube root, divide prime exponents by (3). Step 2: (2^6) becomes (2^2) and (3^3) becomes (3). Step 3: In cube roots, bases remain the same and only exponents change.
Correct answer: A. (2^2 \times 3)
Explanation: Step 1: When taking a square root, halve all prime exponents. Step 2: (2^4) becomes (2^2) and (3^2) becomes (3). Step 3: In square roots, halve the exponent, not the base.
Correct answer: A. 1
Explanation: Step 1: (12=2^2 \times 3). Step 2: From (2^3), (2^2) can be taken only (1) full time, while (3^2) can supply (3) twice. The limiting exponent is for (2), so the answer is (1). Step 3: For a composite divisor, check each required prime separately.
Correct answer: A. 2
Explanation: Step 1: (10=2 \times 5). Step 2: The exponent of (2) is (5) and of (5) is (2), so only (2) complete pairs of (10) can be formed. Step 3: The number of divisions by (10) is decided by the smaller exponent.
Correct answer: B. 72
Explanation: Step 1: For a divisor, its prime exponents must not exceed the available exponents. Step 2: (72=2^3 \times 3^2), which is fully present in the given number. Step 3: To test divisibility, match each prime exponent separately.
Correct answer: A. (2^3 \times 5^3)
Explanation: Step 1: (1000=10^3). Step 2: Since (10=2 \times 5), (10^3=(2 \times 5)^3=2^3 \times 5^3). Step 3: (10^3) is not final prime factorisation because (10) is not prime.
Correct answer: C. (5^2 \times 7^3)
Explanation: Step 1: Look at the exponent of (7) in each option. Step 2: The exponents are (1,2,3,2), and the greatest is (3). Step 3: For comparison, check only the required exponent instead of calculating each value.
Correct answer: C. (2) and (5)
Explanation: Step 1: Compare the exponents. Step 2: The exponent of (2) is (2), of (3) is (1), and of (5) is (2). The greatest exponent is (2), shared by (2) and (5). Step 3: If exponents are equal, more than one prime base may be correct.
Correct answer: C. 225
Explanation: Step 1: (3^2=9) and (5^2=25). Step 2: (9 \times 25=225), so (N=225). Step 3: Remembering squares makes such calculations faster.
Correct answer: C. (2^9)
Explanation: Step 1: Write (512) as (64 \times 8). Step 2: (64=2^6) and (8=2^3), so (512=2^9). Step 3: (8^3) may give the value, but prime factorisation must use base (2).
Correct answer: C. (2) and (3)
Explanation: Step 1: The bases of the first number are (2,3,5). Step 2: The bases of the second number are (2,3,7), so the common primes are (2) and (3). Step 3: For common factors, choose only bases present in both numbers.
Correct answer: A. 2
Explanation: Step 1: In HCF, take the smaller exponent of a common prime. Step 2: The exponents of (2) are (4) and (2), so the smaller exponent is (2). Step 3: For HCF, remember the minimum exponent rule.
Correct answer: B. 3
Explanation: Step 1: In LCM, take the larger exponent of each prime. Step 2: The exponents of (3) are (2) and (3), so the larger exponent is (3). Step 3: For LCM, choose the maximum exponent.
Correct answer: C. (3^4)
Explanation: Step 1: (81=9 \times 9). Step 2: Since (9=3^2), (81=3^2 \times 3^2=3^4). Step 3: (9^2) is not final prime factorisation because (9) is not prime.
Correct answer: C. 8
Explanation: Step 1: An even factor must have exponent of (2) at least (1). Step 2: The exponent of (2) has (2) choices (1,2), while (3) has (2) choices and (5) has (2) choices. Total (2 \times 2 \times 2=8). Step 3: While counting even factors, do not take exponent (0) for (2).
Correct answer: B. 3
Explanation: Step 1: In an odd factor, the exponent of (2) must be (0). Step 2: The exponent of (3) can be (0,1,2), giving (3) odd factors. Step 3: While counting odd factors, remove (2) completely.
Correct answer: A. (\min(a,b)=2)
Explanation: Step 1: Trailing zeros are formed by pairs of (2) and (5). Step 2: The number of pairs equals the smaller of (a) and (b). So (\min(a,b)=2). Step 3: For trailing zeros, use the smaller exponent, not the sum.
Correct answer: A. (1) has no prime factor
Explanation: Step 1: A prime number has exactly two positive factors. Step 2: (1) has only one positive factor, so it is not prime and has no prime factor. Step 3: Avoid the common mistake of treating (1) as prime.
Correct answer: A. Fundamental theorem of arithmetic
Explanation: Step 1: Every integer greater than (1) can be written as a product of prime factors. Step 2: This form is unique apart from order, and this comes from the fundamental theorem of arithmetic. Step 3: Remember uniqueness of prime factorisation with this theorem.
Correct answer: C. (a=3, b=3)
Explanation: Step 1: Write (216) as (8 \times 27). Step 2: (8=2^3) and (27=3^3), so (216=2^3 \times 3^3). Hence (a=3, b=3). Step 3: For unknown exponents, split the number into familiar powers.
