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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. Prime bases and their powers remain fixed except for order
Explanation: Step 1: The theorem says that prime factorisation of a number greater than 1 is fixed. Step 2: The order may change, but the prime bases and their powers do not change. Step 3: Keep only prime factors in the final answer.
Correct answer: A. (2^7\times3^2\times5\times7\times11)
Explanation: Step 1: Write (443520=40320\times11). Step 2: Since (40320=2^7\times3^2\times5\times7), the complete prime form is (2^7\times3^2\times5\times7\times11). Step 3: 40320 is composite, so it should not remain in the final form.
Correct answer: A. 46656
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 (2^6) and (3^6). Step 3: (64\times729=46656), so the answer is 46656.
Correct answer: A. (2^9\times3^8\times5^4\times7^3)
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^9), (3^8), (5^4), and (7^3). Step 3: So the correct form is (2^9\times3^8\times5^4\times7^3).
Correct answer: A. 2520
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: LCM (=3628800\div1440=2520). Step 3: Use this relation directly for exactly two numbers.
Correct answer: A. 7761600
Explanation: Step 1: The product of two numbers equals the product of their HCF and LCM. Step 2: (420\times18480=7761600). Step 3: Apply this formula directly only when exactly two numbers are involved.
Correct answer: A. 6
Explanation: Step 1: (86400=2^7\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. 15
Explanation: Step 1: (105840=2^4\times3^3\times5\times7^2). Step 2: For a perfect square, odd powers of 3 and 5 must be reduced. Step 3: Dividing by (3\times5=15) leaves (2^4\times3^2\times7^2).
Correct answer: A. 1225
Explanation: Step 1: (60480=2^6\times3^3\times5\times7). Step 2: For a perfect cube, every exponent must be a multiple of 3. Step 3: Powers of 5 and 7 are 1, so multiply by (5^2\times7^2=1225).
Correct answer: A. (2^2\times3^6\times7^3)
Explanation: Step 1: Write (1000188=4\times250047). Step 2: (4=2^2) and (250047=3^6\times7^3), so the prime form is (2^2\times3^6\times7^3). Step 3: Treat 729 and 343 as powers of prime bases and keep prime bases in the final answer.
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 other powers are even. Step 3: Multiplying by (2\times5=10) makes all powers even.
Correct answer: A. (2\times3\times5\times7)
Explanation: Step 1: For a perfect cube, exponents must be multiples of 3. Step 2: We must make 11 to 12, 8 to 9, 5 to 6, and 2 to 3. Step 3: Therefore, the smallest multiplier is (2\times3\times5\times7).
Correct answer: C. 16
Explanation: Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 3 in (a) is 7 and in (b) is 9. Step 3: In (ab), the power of 3 is (7+9=16).
Correct answer: C. 14
Explanation: Step 1: In (xy), powers with the same base 5 are added. Step 2: The power of 5 in (x) is 8 and in (y) is 6. Step 3: The total power is (8+6=14).
Correct answer: A. (2^{10}\times3^9)
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^{10}) and (3^9). Step 3: Therefore, the correct form is (2^{10}\times3^9).
Correct answer: A. (2^{13}\times3^{11}\times5^5\times7^4)
Explanation: Step 1: LCM uses the highest powers of all prime factors. Step 2: The highest powers are (2^{13}), (3^{11}), (5^5), and (7^4). Step 3: So the correct form is (2^{13}\times3^{11}\times5^5\times7^4).
Correct answer: A. 3960
Explanation: Step 1: Product of the two numbers is (180\times27720=4989600). Step 2: One number is 1260, so the other number is (4989600\div1260=3960). Step 3: As a check, the HCF of 1260 and 3960 is 180.
Correct answer: A. 4320
Explanation: Step 1: Product of the two numbers is (216\times30240=6531840). Step 2: The other number is (6531840\div1512=4320). Step 3: To check, the HCF of 1512 and 4320 is 216.
Correct answer: B. 5
Explanation: Step 1: Write (90720=2592\times35). Step 2: (2592=2^5\times3^4) and (35=5\times7), so (p=2^5\times3^4\times5\times7). Step 3: Comparing gives (a=5).
Correct answer: C. 4
Explanation: Step 1: (3628800=2^8\times3^4\times5^2\times7). Step 2: The given power of 2 does not fully match, but the power of 3 is clearly 4. Step 3: Therefore, (b=4).
Correct answer: C. 27
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (2^{10}) gives 10, (3^8) gives 8, (5^7) gives 7, and (19^2) gives 2 factors. Step 3: Total (10+8+7+2=27), so the answer is 27.
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 17. 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. 12673
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 12673.
Correct answer: A. (2^{17})
Explanation: Step 1: Divide 131072 repeatedly by 2. Step 2: (131072=2^{17}). Step 3: (4^9) and (16^5) are not prime factorisations because 4 and 16 are composite.
