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
अंकगणित का मौलिक प्रमेय
In this Class 10 Mathematics topic from the chapter Real Numbers, students learn that every integer greater than 1 can be expressed as a product of prime numbers, and that this prime factorisation is unique apart from the order of the factors. They practise finding prime factors and use the theorem to understand and determine the HCF and LCM of numbers. The topic builds clear reasoning about the structure of whole numbers and supports later work with divisibility and number relationships.
Correct answer: A. Prime factorisation
Explanation: Step 1: This theorem is related to writing numbers as products of prime factors. Step 2: Every positive integer greater than 1 can be written as a product of primes. Step 3: In this chapter, connect it with factorisation and HCF.
Correct answer: A. (2^4\times3)
Explanation: Step 1: 48 can be written as (16\times3). Step 2: (16=2^4), so (48=2^4\times3). Step 3: Do not leave composite factors like 4 or 12 in the final prime factorisation.
Correct answer: A. (2^3\times3^2)
Explanation: Step 1: Write (72=8\times9). Step 2: (8=2^3) and (9=3^2), so (72=2^3\times3^2). Step 3: 8 and 9 are composite, so they are not the final prime form.
Correct answer: A. The number remains the same
Explanation: Step 1: Changing the order in multiplication does not change the product. Step 2: For example, (2\times3\times5) and (5\times3\times2) both give 30. Step 3: In prime factorisation, the factors matter, not their order.
Correct answer: B. 5
Explanation: Step 1: Write (96=32\times3). Step 2: (32=2^5), so (96=2^5\times3). Step 3: Count repeated prime factors to write their powers.
Correct answer: C. 180
Explanation: Step 1: First evaluate the powers: (2^2=4) and (3^2=9). Step 2: (4\times9\times5=180), so the number is 180. Step 3: Simplify powers first, then multiply.
Correct answer: A. (3^2\times7)
Explanation: Step 1: Write (63=9\times7). Step 2: (9=3^2), so (63=3^2\times7). Step 3: 9 and 21 are composite, so do not keep them in the final form.
Correct answer: A. 31
Explanation: Step 1: A prime number has exactly two positive factors. Step 2: 31 is divisible only by 1 and 31, while 27, 35, and 45 are composite. Step 3: For small numbers, checking divisibility by 2, 3, 5, and 7 is useful.
Correct answer: C. 77
Explanation: Step 1: A composite number has more than two positive factors. Step 2: (77=7\times11), so it is composite. Step 3: If a number can be written as a product of two smaller primes, it is composite.
Correct answer: A. (3^3\times5)
Explanation: Step 1: Write (135=27\times5). Step 2: (27=3^3), so (135=3^3\times5). Step 3: 9 and 15 may form the product, but they are not prime factorisation.
Correct answer: B. 3
Explanation: Step 1: Look at distinct prime factors, not their powers. Step 2: In (168=2^3\times3\times7), the distinct primes are 2, 3, and 7. Step 3: Even if a power is large, count that prime once in the distinct list.
Correct answer: A. (5^3)
Explanation: Step 1: Write (125=5\times25). Step 2: (25=5\times5), so (125=5^3). Step 3: 25 is composite, so (25\times5) is not the final prime form.
Correct answer: A. 12
Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller power of (2) is 2 and of (3) is 1, so (2^2\times3=12). Step 3: Use smaller powers for HCF.
Correct answer: C. 72
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest power of (2) is 3 and of (3) is 2, so (2^3\times3^2=72). Step 3: Use highest powers for LCM.
Correct answer: A. (2^3\times3^3)
Explanation: Step 1: Write (216=8\times27). Step 2: (8=2^3) and (27=3^3), so (216=2^3\times3^3). Step 3: (6^3) gives the value, but 6 is not prime.
Correct answer: C. 80
Explanation: Step 1: First find (2^4=16). Step 2: (16\times5=80), so the number is 80. Step 3: In a form with powers, it is easier to evaluate the power first.
Correct answer: A. All factors should be prime numbers
Explanation: Step 1: In prime factorisation, a number is written as a product of prime numbers. Step 2: So the final form should contain only prime factors. Step 3: If any composite factor remains, factorise it further.
Correct answer: A. (2\times7^2)
Explanation: Step 1: Write (98=2\times49). Step 2: (49=7^2), so (98=2\times7^2). Step 3: 14 is composite, so (7\times14) is not the final prime factorisation.
Correct answer: A. 2 and 11
Explanation: Step 1: Write (88=8\times11). Step 2: (8=2^3), so (88=2^3\times11). Step 3: The distinct prime factors are 2 and 11.
Correct answer: A. Only 1
Explanation: Step 1: Co-prime numbers have no common factor except 1. Step 2: Therefore, their only common positive factor is 1. Step 3: To identify co-prime numbers, look for common prime factors.
Correct answer: A. 9 and 16
Explanation: Step 1: (9=3^2) and (16=2^4). Step 2: They have no common prime factor, so they are co-prime. Step 3: If a common prime factor is found, the pair is not co-prime.
Correct answer: A. 30
Explanation: Step 1: The common prime factors are 2, 3, and 5. Step 2: The smaller powers are (2^1), (3^1), and (5^1), so the HCF is (2\times3\times5=30). Step 3: Take smaller powers for HCF.
Correct answer: C. 180
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^2), (3^2), and (5^1), so (4\times9\times5=180). Step 3: Use highest powers for LCM.
Correct answer: A. (2^2\times3^2\times7)
Explanation: Step 1: Write (252=4\times63). Step 2: (4=2^2) and (63=3^2\times7), so (252=2^2\times3^2\times7). Step 3: Do not keep composite factors in the final answer.
Correct answer: C. 63
Explanation: Step 1: Find (3^2=9). Step 2: (9\times7=63), so the number is 63. Step 3: To get the number from prime factorisation, multiply all factors.
