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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. As a product of prime numbers
Explanation: Step 1: This theorem is about prime factorisation. Step 2: Every composite number can be written as a product of prime numbers. Step 3: In exams, remember this theorem through factorisation.
Correct answer: A. (2^2\times3^2)
Explanation: Step 1: Break 36 into small prime factors. Step 2: (36=2\times2\times3\times3=2^2\times3^2). Step 3: In the final answer, keep only prime numbers as factors.
Correct answer: A. (2^2\times3\times5)
Explanation: Step 1: Write 60 as (2\times30), then (30=2\times3\times5). Step 2: So (60=2^2\times3\times5). Step 3: (4\times15) gives the product, but it is not prime factorisation.
Correct answer: A. Prime factors remain the same even if their order changes
Explanation: Step 1: Uniqueness means the set of prime factors is fixed. Step 2: The order may change, such as (2\times3) and (3\times2), but the factors remain the same. Step 3: Do not get confused by the order of factors.
Correct answer: B. 2
Explanation: Step 1: (84=2\times42=2\times2\times21). Step 2: So (84=2^2\times3\times7), where the power of 2 is 2. Step 3: To find the power, count repeated prime factors.
Correct answer: A. 72
Explanation: Step 1: First find the powers: (2^3=8) and (3^2=9). Step 2: (8\times9=72), so the number is 72. Step 3: Calculate powers first, then multiply.
Correct answer: A. (3^2\times5)
Explanation: Step 1: (45=9\times5) and (9=3\times3). Step 2: Therefore, (45=3^2\times5). Step 3: Composite factors like 9 or 15 are not kept in final prime factorisation.
Correct answer: A. 29
Explanation: Step 1: A prime number has exactly two positive factors, 1 and itself. Step 2: 29 is divisible only by 1 and 29, while the other numbers are composite. Step 3: For small numbers, checking divisibility by 2, 3, 5, and 7 is useful.
Correct answer: C. 49
Explanation: Step 1: A composite number has more than two positive factors. Step 2: (49=7\times7), so it has a factor 7 besides 1 and 49. Step 3: Recognising perfect squares helps in finding composite numbers.
Correct answer: A. (2\times3^2\times5)
Explanation: Step 1: Write (90=9\times10). Step 2: (9=3^2) and (10=2\times5), so (90=2\times3^2\times5). Step 3: In the final form, all factors must be prime.
Correct answer: B. 2
Explanation: Step 1: Look at distinct prime factors, not their powers. Step 2: In (72=2^3\times3^2), the distinct primes are 2 and 3. Step 3: The number of distinct prime factors and total prime factors are different.
Correct answer: A. (2^2\times5^2)
Explanation: Step 1: Write (100=10\times10). Step 2: Each 10 is (2\times5), so (100=2^2\times5^2). Step 3: 4, 10, and 25 are composite, so do not keep them in the final prime form.
Correct answer: A. 6
Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller power of (2) is 1 and of (3) is 1, so (2\times3=6). Step 3: Use smaller powers for HCF.
Correct answer: D. 36
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The higher power of (2) is 2 and of (3) is 2, so (2^2\times3^2=36). Step 3: Use higher powers for LCM.
Correct answer: A. (2^4\times3^2)
Explanation: Step 1: Write (144=16\times9). Step 2: (16=2^4) and (9=3^2), so (144=2^4\times3^2). Step 3: 12 is composite, so (12\times12) is not prime factorisation.
Correct answer: B. 45
Explanation: Step 1: First find (3^2=9). Step 2: (9\times5=45), so the number is 45. Step 3: In factorisation with powers, evaluate the power first.
Correct answer: A. All factors should be prime
Explanation: Step 1: Prime factorisation means writing a number as a product of prime numbers. Step 2: So the final factors should be primes such as 2, 3, 5, and 7. Step 3: If a composite factor remains, factorise it further.
Correct answer: A. (3\times5^2)
Explanation: Step 1: Write (75=3\times25). Step 2: (25=5\times5=5^2), so (75=3\times5^2). Step 3: 15 and 25 are composite, so do not keep them in the final form.
Correct answer: A. 2 and 7
Explanation: Step 1: Write (56=8\times7). Step 2: (8=2^3), so (56=2^3\times7). Step 3: The distinct prime factors are 2 and 7.
Correct answer: A. 1
Explanation: Step 1: Co-prime numbers have no common factor except 1. Step 2: Therefore, their HCF is 1. Step 3: To identify co-prime numbers, check common prime factors.
Correct answer: A. 8 and 15
Explanation: Step 1: Co-prime numbers have no common prime factor. Step 2: (8=2^3) and (15=3\times5), so they have no common prime factor. Step 3: If a common factor appears, the pair is not co-prime.
Correct answer: A. 10
Explanation: Step 1: The common prime factors are 2 and 5. Step 2: The smaller powers are (2^1) and (5^1), so the HCF is (2\times5=10). Step 3: Remember to take smaller powers for HCF.
Correct answer: C. 200
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^3) and (5^2), so (8\times25=200). Step 3: LCM uses the highest powers.
Correct answer: A. (2^3\times3\times5)
Explanation: Step 1: Write (120=12\times10). Step 2: (12=2^2\times3) and (10=2\times5), so (120=2^3\times3\times5). Step 3: Write repeated prime factors using powers.
Correct answer: C. 28
Explanation: Step 1: Find (2^2=4). Step 2: (4\times7=28), so the number is 28. Step 3: To get the number from prime factorisation, simplify powers first.
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