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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. Except only for the order of factors
Explanation: Step 1: The theorem gives certainty of prime factorisation for numbers greater than 1. Step 2: Prime factors and their powers remain fixed; only their order may change. Step 3: In the final answer, do not leave composite factors.
Correct answer: A. (2^6\times3^2\times5\times7\times11)
Explanation: Step 1: Write (221760=20160\times11). Step 2: Since (20160=2^6\times3^2\times5\times7), the complete prime form is (2^6\times3^2\times5\times7\times11). Step 3: 20160 is composite, so do not keep it in the final answer.
Correct answer: A. 7776
Explanation: Step 1: For HCF, take the smaller powers of common prime factors only. Step 2: The common factors are 2 and 3, with smaller powers (2^5) and (3^5). Step 3: (32\times243=7776), so the answer is 7776.
Correct answer: A. 38482790400
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^8), (3^7), (5^2), and (7^3). Step 3: Their product is 38482790400, so that is the answer.
Correct answer: A. 2520
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: LCM (=1814400\div720=2520). Step 3: Use this relation directly only for two numbers.
Correct answer: A. 1940400
Explanation: Step 1: The product of two numbers equals the product of their HCF and LCM. Step 2: (210\times9240=1940400). Step 3: Apply this formula directly only when exactly two numbers are involved.
Correct answer: A. 6
Explanation: Step 1: (21600=216\times100=2^5\times3^3\times5^2). Step 2: For a perfect square, all exponents must be even, but powers of 2 and 3 are odd. Step 3: Multiplying by (2\times3=6) makes all powers even.
Correct answer: A. 30
Explanation: Step 1: (52920=2^3\times3^3\times5\times7^2). Step 2: For a perfect square, odd powers of 2, 3, and 5 must be reduced. Step 3: Dividing by (2\times3\times5=30) leaves (2^2\times3^2\times7^2).
Correct answer: A. 2450
Explanation: Step 1: (30240=2^5\times3^3\times5\times7). Step 2: For a perfect cube, exponents must be multiples of 3. Step 3: We need (2), (5^2), and (7^2), so the smallest multiplier is (2\times25\times49=2450).
Correct answer: A. (2\times3^3\times7^3\times13)
Explanation: Step 1: Write (500094=2\times250047). Step 2: (250047=3^3\times7^3\times13), so the full form is (2\times3^3\times7^3\times13). Step 3: Composite bases should not remain in final prime form.
Correct answer: A. 10
Explanation: Step 1: For a perfect square, every exponent must be even. Step 2: Powers of 2 and 5 are odd, while the others are even. Step 3: Multiplying by (2\times5=10) makes all powers even.
Correct answer: A. (2^2\times3\times5\times7^2)
Explanation: Step 1: For a perfect cube, every exponent must be a multiple of 3. Step 2: We must make 10 to 12, 8 to 9, 5 to 6, and 1 to 3. Step 3: So the smallest multiplier is (2^2\times3\times5\times7^2).
Correct answer: C. 14
Explanation: Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 3 in (a) is 6 and in (b) is 8. Step 3: In (ab), the power of 3 is (6+8=14).
Correct answer: C. 12
Explanation: Step 1: In (xy), powers with the same base 5 are added. Step 2: The power of 5 in (x) is 7 and in (y) is 5. Step 3: The total power is (7+5=12).
Correct answer: A. (2^9\times3^8)
Explanation: Step 1: HCF uses the smaller powers of common prime factors only. Step 2: The common factors are 2 and 3, with smaller powers (2^9) and (3^8). Step 3: So the correct form is (2^9\times3^8).
Correct answer: A. (2^{12}\times3^{10}\times5^4\times7^5)
Explanation: Step 1: LCM uses the highest powers of all prime factors. Step 2: The highest powers are (2^{12}), (3^{10}), (5^4), and (7^5). Step 3: So the correct form is (2^{12}\times3^{10}\times5^4\times7^5).
Correct answer: A. 1320
Explanation: Step 1: Product of the two numbers is (120\times9240=1108800). Step 2: One number is 840, so the other is (1108800\div840=1320). Step 3: As a check, the HCF of 840 and 1320 is 120.
Correct answer: A. 2016
Explanation: Step 1: Product of the two numbers is (144\times10080=1451520). Step 2: The other number is (1451520\div720=2016). Step 3: To check, the HCF of 720 and 2016 is 144.
Correct answer: B. 4
Explanation: Step 1: Write (136080=3888\times35). Step 2: (3888=2^4\times3^5) and (35=5\times7), so (p=2^4\times3^5\times5\times7). Step 3: Comparing gives (a=4).
Correct answer: B. 4
Explanation: Step 1: Write (1814400=18144\times100). Step 2: (18144=2^5\times3^4\times7) and (100=2^2\times5^2), so the actual form is (2^7\times3^4\times5^2\times7). Step 3: The power of 3 is (b=4).
Correct answer: C. 24
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (2^9) gives 9, (3^7) gives 7, (5^6) gives 6, and (17^2) gives 2 factors. Step 3: Total (9+7+6+2=24), so the answer is 24.
Correct answer: A. 5
Explanation: Step 1: When counting distinct prime factors, exponents are not added. Step 2: The prime bases are 2, 3, 5, 7, and 13. Step 3: Therefore, the number of distinct prime factors is 5.
Correct answer: A. They are co-prime
Explanation: Step 1: The prime factors of the first number are 2, 3, and 13. Step 2: The prime factors of the second number are 5, 7, and 11. Step 3: There is no common prime factor, so they are co-prime.
Correct answer: D. 7429
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 7429.
Correct answer: A. (2^{16})
Explanation: Step 1: Divide 65536 repeatedly by 2. Step 2: (65536=2^{16}). Step 3: (4^8) and (16^4) can give the value, but 4 and 16 are not prime.
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