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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. (2^4\times3^2\times5)
Explanation: Step 1: Write (720=72\times10). Step 2: (72=2^3\times3^2) and (10=2\times5), so (720=2^4\times3^2\times5). Step 3: Do not keep composite factors in the final answer.
Correct answer: A. (2^2\times3\times7\times11)
Explanation: Step 1: Write (924=84\times11). Step 2: (84=2^2\times3\times7), so (924=2^2\times3\times7\times11). Step 3: 84 is composite, so do not leave it in the final prime form.
Correct answer: A. (2^3\times3\times7^2)
Explanation: Step 1: Write (1176=24\times49). Step 2: (24=2^3\times3) and (49=7^2), so (1176=2^3\times3\times7^2). Step 3: 24 and 49 are composite, so write prime powers in the final form.
Correct answer: A. 120
Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller powers are (2^3), (3^1), and (5^1). Step 3: (8\times3\times5=120), so the answer is 120.
Correct answer: B. 50400
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^5), (3^2), (5^2), and (7). Step 3: (32\times9\times25\times7=50400), so the answer is 50400.
Correct answer: C. 120
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: LCM (=2520\div21=120). Step 3: Use this formula directly only for two numbers.
Correct answer: A. 8640
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (24\times360=8640). Step 3: Apply the HCF-LCM relation carefully.
Correct answer: C. 1008
Explanation: Step 1: First calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times7=1008). Step 3: Simplify powers first while finding the number.
Correct answer: B. 3
Explanation: Step 1: Write (2520=252\times10). Step 2: (252=2^2\times3^2\times7) and (10=2\times5), so (2520=2^3\times3^2\times5\times7). Step 3: The number of times 2 appears is its power.
Correct answer: A. (3^2\times5^2\times7)
Explanation: Step 1: Write (1575=15\times105). Step 2: (15=3\times5) and (105=3\times5\times7), so (1575=3^2\times5^2\times7). Step 3: Do not keep composite factors in the final answer.
Correct answer: C. 720
Explanation: Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5=720). Step 3: To get the number from prime factorisation, multiply all factors.
Correct answer: C. 4
Explanation: Step 1: Look at distinct prime factors, not powers. Step 2: In (m), the distinct primes are 2, 3, 7, and 11. Step 3: Count 2 and 3 only once each despite their powers.
Correct answer: A. They are co-prime
Explanation: Step 1: (27=3^3) and (64=2^6). Step 2: They have no common prime factor. Step 3: Therefore, they are co-prime and their HCF is 1.
Correct answer: C. 391
Explanation: Step 1: Co-prime numbers have HCF 1. Step 2: For two numbers, product (=) HCF (\times) LCM. Step 3: Therefore, the LCM will be 391.
Correct answer: A. (2^3\times3\times13)
Explanation: Step 1: In prime factorisation, every factor must be prime. Step 2: (2), (3), and (13) are prime, so (2^3\times3\times13) is correct. Step 3: 8, 12, and 24 are composite, so they cannot remain in the final form.
Correct answer: B. 24
Explanation: Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^3) and (3^1). Step 3: (8\times3=24), so the HCF is 24.
Correct answer: B. 1008
Explanation: Step 1: For LCM, take the highest powers. Step 2: The highest powers are (2^4), (3^2), and (7). Step 3: (16\times9\times7=1008), so the answer is 1008.
Correct answer: A. (2\times3^3\times5\times7)
Explanation: Step 1: Write (1890=189\times10). Step 2: (189=3^3\times7) and (10=2\times5). Step 3: Therefore, (1890=2\times3^3\times5\times7).
Correct answer: B. 576
Explanation: Step 1: Calculate (2^6=64) and (3^2=9). Step 2: (64\times9=576). Step 3: In a form with powers, it is easier to evaluate powers first.
Correct answer: C. 9
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (2^5) gives 5, (3^3) gives 3, and (7) gives 1 factor. Step 3: Total (5+3+1=9), so the answer is 9.
Correct answer: B. 2
Explanation: Step 1: Prime factorise 180. Step 2: (180=18\times10=2^2\times3^2\times5). Step 3: Comparing with the given form gives (a=2).
Correct answer: C. 3
Explanation: Step 1: Write (1512=8\times189). Step 2: (8=2^3) and (189=3^3\times7), so (1512=2^3\times3^3\times7). Step 3: Comparing gives (b=3).
Correct answer: A. They are co-prime
Explanation: Step 1: The prime factors of the first number are 2 and 7. Step 2: The prime factors of the second number are 3 and 5. Step 3: There is no common prime factor, so they are co-prime.
Correct answer: D. 1260
Explanation: Step 1: (28=2^2\times7) and (45=3^2\times5), so they are co-prime. Step 2: The LCM of co-prime numbers equals their product. Step 3: (28\times45=1260), so the answer is 1260.
Correct answer: A. (2^{11})
Explanation: Step 1: Divide 2048 repeatedly by 2. Step 2: (2048=2^{11}). Step 3: 4 and 32 are composite, so do not use them in the final prime form.
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