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. (2^{10})
Explanation: Step 1: Divide 1024 repeatedly by 2. Step 2: (1024=2^{10}). Step 3: (4^5) and (32^2) can give the value, but 4 and 32 are not prime.
Correct answer: B. 2025
Explanation: Step 1: Calculate (3^4=81) and (5^2=25). Step 2: (81\times25=2025). Step 3: In questions with powers, simplify the powers first.
Correct answer: A. (3^3\times5^2)
Explanation: Step 1: Write (675=27\times25). Step 2: (27=3^3) and (25=5^2), so (675=3^3\times5^2). Step 3: 27 and 25 are composite, so write their prime powers in the final form.
Correct answer: C. 6
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (2^2) gives 2, (3^3) gives 3, and (5) gives 1 factor. Step 3: Total (2+3+1=6), so the answer is 6.
Correct answer: A. 72
Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller powers are (2^3) and (3^2). Step 3: (2^3\times3^2=8\times9=72), so the answer is 72.
Correct answer: B. 2592
Explanation: Step 1: For LCM, take the highest powers. Step 2: The highest powers are (2^5) and (3^4). Step 3: (32\times81=2592), so the answer is 2592.
Correct answer: A. 3, 5, 7 and 11
Explanation: Step 1: Write (1155=105\times11). Step 2: (105=3\times5\times7), so (1155=3\times5\times7\times11). Step 3: The distinct prime factors are 3, 5, 7, and 11.
Correct answer: B. 8820
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (21\times420=8820). Step 3: Use the HCF-LCM relation in the correct situation.
Correct answer: A. (3^4\times5^2)
Explanation: Step 1: Write (2025=45\times45). Step 2: (45=3^2\times5), so (2025=3^4\times5^2). Step 3: In perfect squares, the powers of the base factors get doubled.
Correct answer: A. 1 is neither prime nor composite
Explanation: Step 1: A prime number must have exactly two positive factors. Step 2: 1 has only one positive factor, so it is not prime. Step 3: A composite number needs more than two factors, so 1 is not composite either.
Correct answer: B. 20
Explanation: Step 1: The common prime factors are 2 and 5. Step 2: The smaller powers are (2^2) and (5^1). Step 3: (2^2\times5=4\times5=20), so the HCF is 20.
Correct answer: B. 4200
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^3), (3), (5^2), and (7). Step 3: (8\times3\times25\times7=4200), so the answer is 4200.
Correct answer: A. (2^3\times3\times7\times11)
Explanation: Step 1: Write (1848=168\times11). Step 2: (168=2^3\times3\times7), so (1848=2^3\times3\times7\times11). Step 3: 168 is composite, so do not keep it in the final form.
Correct answer: A. Every composite number has a unique prime factorisation except for order
Explanation: Step 1: The main idea of the theorem is prime factorisation. Step 2: Every composite number greater than 1 can be written in a fixed way as prime factors. Step 3: The order may change, but the prime factors do not change.
Correct answer: B. 4800
Explanation: Step 1: Calculate (2^6=64) and (5^2=25). Step 2: (64\times3\times25=4800). Step 3: Simplifying powers first makes multiplication easier.
Correct answer: A. 1
Explanation: Step 1: Having no common prime factor means the numbers are co-prime. Step 2: Co-prime numbers have HCF 1. Step 3: Prime factorisation helps identify co-primality quickly.
Correct answer: C. 336
Explanation: Step 1: Calculate (2^4=16). Step 2: (16\times3\times7=336). Step 3: To get the number from prime factorisation, multiply all factors.
Correct answer: C. 3
Explanation: Step 1: Write (1080=108\times10). Step 2: (108=2^2\times3^3) and (10=2\times5), so (1080=2^3\times3^3\times5). Step 3: Comparing with the given form gives (a=3).
Correct answer: C. 3
Explanation: Step 1: In multiplication, powers with the same base are added. Step 2: The power of 3 in (a) is 2 and in (b) is 1. Step 3: In (ab), the power of 3 will be (2+1=3).
Correct answer: C. 5
Explanation: Step 1: While multiplying, add the exponents of the same prime base. Step 2: The power of 5 in (a) is 2 and in (b) is 3. Step 3: In (ab), the power of 5 will be (2+3=5).
Correct answer: A. 10
Explanation: Step 1: (2^3) and (5^2) contain both factors 2 and 5. Step 2: The product of 2 and 5 is 10, so the number must be divisible by 10. Step 3: Divisibility can be quickly identified from prime factors.
Correct answer: A. (3^3\times7^2)
Explanation: Step 1: Write (1323=27\times49). Step 2: (27=3^3) and (49=7^2), so (1323=3^3\times7^2). Step 3: 27 and 49 are composite, so write prime powers in the final form.
Correct answer: C. 5
Explanation: Step 1: In (xy), the powers of the same base 2 are added. Step 2: The power of 2 in (x) is 3 and in (y) is 2. Step 3: So the power of 2 in (xy) is (3+2=5).
Correct answer: A. 60
Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller powers are (2^2), (3^1), and (5^1). Step 3: (4\times3\times5=60), so the HCF is 60.
Correct answer: A. Prime factorisation is unique except for order
Explanation: Step 1: The theorem tells us that prime factorisation is fixed. Step 2: The order of factors may change but the prime factors remain the same. Step 3: In exams, do not treat changed order as a new factorisation.
Student feedback
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesNo published reviews yet.
