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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. The set of prime factors remains fixed except for order
Explanation: Step 1: This theorem says that the prime factorisation of a number greater than 1 is fixed. Step 2: The order may change, but the prime factors do not change. Step 3: In exams, do not treat changed order as a new factorisation.
Correct answer: A. (2^3\times3^3\times5\times7\times11)
Explanation: Step 1: Write (83160=7560\times11). Step 2: Since (7560=2^3\times3^3\times5\times7), the full factorisation is (2^3\times3^3\times5\times7\times11). Step 3: Do not leave a composite factor like 7560 in the final answer.
Correct answer: A. 3888
Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The common factors are 2 and 3, with smaller powers (2^4) and (3^5). Step 3: (2^4\times3^5=16\times243=3888), so the answer is 3888.
Correct answer: A. 857304000
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^6), (3^7), (5^3), and (7^2). Step 3: (64\times2187\times125\times49=857304000), so the answer is 857304000.
Correct answer: A. 2520
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: LCM (=362880\div144=2520). Step 3: Use this relation directly only for two numbers.
Correct answer: A. 291060
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (63\times4620=291060). Step 3: In such questions, directly multiply the two given values.
Correct answer: A. 3
Explanation: Step 1: (10800=108\times100=2^4\times3^3\times5^2). Step 2: For a perfect square, all exponents must be even, but the exponent of 3 is 3. Step 3: Multiplying by 3 makes the exponent 4, so the smallest number is 3.
Correct answer: D. 30
Explanation: Step 1: (13230=2\times3^3\times5\times7^2). Step 2: For a perfect square, all exponents must be even, so the odd powers of 2, 3, and 5 must be reduced. Step 3: Dividing by (2\times3\times5=30) leaves (3^2\times7^2), a perfect square.
Correct answer: A. 1225
Explanation: Step 1: (7560=2^3\times3^3\times5\times7). Step 2: For a perfect cube, each exponent must be a multiple of 3. Step 3: The powers of 5 and 7 are 1, so multiply by (5^2\times7^2=1225).
Correct answer: A. (2^3\times3^3\times7^3)
Explanation: Step 1: Recognise (74088=42^3). Step 2: Since (42=2\times3\times7), (42^3=2^3\times3^3\times7^3). Step 3: 42 is composite, so (42^3) is not the final prime form.
Correct answer: A. 6
Explanation: Step 1: For a perfect square, all exponents must be even. Step 2: The exponents of 2 and 3 are odd, while the others are even. Step 3: Multiplying by (2\times3=6) makes all exponents even.
Correct answer: A. (2^2\times3\times5^2)
Explanation: Step 1: For a perfect cube, every exponent must be a multiple of 3. Step 2: We must make (2^7), (3^5), and (5^4) into powers 9, 6, and 6. Step 3: So the smallest multiplier is (2^2\times3\times5^2).
Correct answer: C. 10
Explanation: Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 3 in (a) is 4 and in (b) is 6. Step 3: In (ab), the power of 3 will be (4+6=10).
Correct answer: C. 7
Explanation: Step 1: In (xy), powers with the same base 5 are added. Step 2: The power of 5 in (x) is 5 and in (y) is 2. Step 3: The total power is (5+2=7).
Correct answer: A. (2^7\times3^6)
Explanation: Step 1: HCF uses only common prime factors. Step 2: The common factors are 2 and 3, with smaller powers (2^7) and (3^6). Step 3: Therefore, the HCF is (2^7\times3^6).
Correct answer: A. (2^{10}\times3^8\times5^2\times7^3)
Explanation: Step 1: LCM uses the highest powers of all prime factors. Step 2: The highest powers are (2^{10}), (3^8), (5^2), and (7^3). Step 3: So the correct form is (2^{10}\times3^8\times5^2\times7^3).
Correct answer: A. 672
Explanation: Step 1: Product of the two numbers is (48\times3360=161280). Step 2: One number is 240, so the other is (161280\div240=672). Step 3: As a check, the HCF of 240 and 672 is 48.
Correct answer: A. 504
Explanation: Step 1: Product of the two numbers is (72\times2520=181440). Step 2: The other number is (181440\div360=504). Step 3: To check, the HCF of 360 and 504 is 72.
Correct answer: B. 3
Explanation: Step 1: Prime factorise 7560. Step 2: (7560=756\times10=2^3\times3^3\times5\times7). Step 3: Comparing with the given form gives (a=3).
Correct answer: C. 3
Explanation: Step 1: Write (15120=16\times945). Step 2: (945=3^3\times5\times7), so (15120=2^4\times3^3\times5\times7). Step 3: Comparing gives (b=3).
Correct answer: C. 17
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (2^7) gives 7, (3^5) gives 5, (5^4) gives 4, and 11 gives 1 factor. Step 3: Total (7+5+4+1=17), so the answer is 17.
Correct answer: A. 4
Explanation: Step 1: When counting distinct prime factors, do not add exponents. Step 2: The prime factors here are 2, 3, 5, and 11. Step 3: Therefore, the number of distinct prime factors is 4.
Correct answer: A. They are co-prime
Explanation: Step 1: The first number has prime factors 2 and 3. Step 2: The second number has prime factors 5 and 7. Step 3: There is no common prime factor, so they are co-prime.
Correct answer: D. 2431
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 2431.
Correct answer: A. (2^{14})
Explanation: Step 1: Divide 16384 repeatedly by 2. Step 2: (16384=2^{14}). Step 3: 4 and 16 are composite, so they should not be written in the final prime form.
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