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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: C. 1620
Explanation: Step 1: Calculate (2^2=4) and (3^4=81). Step 2: (4\times81\times5=1620). Step 3: Simplifying powers first makes multiplication easier.
Correct answer: A. (3^2\times5^3)
Explanation: Step 1: Write (1125=45\times25). Step 2: (45=3^2\times5) and (25=5^2), so (1125=3^2\times5^3). Step 3: 45 and 25 are composite, so write their prime powers in the final form.
Correct answer: C. 7
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (2^3) gives 3, (3^2) gives 2, and (7^2) gives 2 factors. Step 3: Total (3+2+2=7), so the answer is 7.
Correct answer: A. 108
Explanation: Step 1: For HCF, take smaller powers. Step 2: The smaller power of 2 is 2 and of 3 is 3. Step 3: (2^2\times3^3=4\times27=108), so the answer is 108.
Correct answer: B. 3888
Explanation: Step 1: For LCM, take the highest powers. Step 2: The highest powers are (2^4) and (3^5). Step 3: (16\times243=3888), so the answer is 3888.
Correct answer: A. 3, 5, 7 and 13
Explanation: Step 1: Write (1365=105\times13). Step 2: (105=3\times5\times7), so (1365=3\times5\times7\times13). Step 3: The distinct prime factors are 3, 5, 7, and 13.
Correct answer: B. 11760
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (28\times420=11760). Step 3: In this relation, multiply the two given values directly.
Correct answer: B. 14
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: HCF (=5040\div360=14). Step 3: After finding the answer, you can check by multiplying.
Correct answer: A. (3^3\times5^3)
Explanation: Step 1: Write (3375=27\times125). Step 2: (27=3^3) and (125=5^3), so (3375=3^3\times5^3). Step 3: 27 and 125 are composite, so keep prime powers in the final form.
Correct answer: A. 1 is neither prime nor composite
Explanation: Step 1: A prime number must have exactly two positive factors. Step 2: 1 has only one positive factor, so it is not prime. Step 3: A composite number needs more than two factors, so 1 is not composite either.
Correct answer: A. 12
Explanation: Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^2) and (3^1). Step 3: (2^2\times3=4\times3=12), so the HCF is 12.
Correct answer: B. 17640
Explanation: Step 1: For LCM, take the highest powers. Step 2: The highest powers are (2^3), (3^2), (5), and (7^2). Step 3: (8\times9\times5\times49=17640), so the answer is 17640.
Correct answer: A. (2^2\times3^2\times7\times11)
Explanation: Step 1: Write (2772=252\times11). Step 2: (252=2^2\times3^2\times7), so (2772=2^2\times3^2\times7\times11). Step 3: 252 is composite, so it should not remain in the final form.
Correct answer: A. Every number greater than 1 has a unique prime factorisation except for order
Explanation: Step 1: The theorem is about prime factorisation of numbers greater than 1. Step 2: This factorisation is unique except for order. Step 3: Do not mix it with separate rules of HCF or co-primality.
Correct answer: B. 7200
Explanation: Step 1: Calculate (2^5=32), (3^2=9), and (5^2=25). Step 2: (32\times9\times25=7200). Step 3: In larger multiplication, simplify powers first.
Correct answer: A. Co-prime numbers
Explanation: Step 1: Having no common prime factor means their only common factor is 1. Step 2: Such numbers are called co-prime numbers. Step 3: Prime factorisation helps identify co-primality quickly.
Correct answer: B. 672
Explanation: Step 1: Calculate (2^5=32). Step 2: (32\times3\times7=672). Step 3: To get the number from prime factorisation, multiply all factors.
Correct answer: C. 4
Explanation: Step 1: Write (6480=648\times10). Step 2: (648=2^3\times3^4) and (10=2\times5), so (6480=2^4\times3^4\times5). Step 3: Comparing gives (a=4).
Correct answer: C. 3
Explanation: Step 1: While multiplying, exponents of the same prime base are added. Step 2: The power of 5 in (a) is 1 and in (b) is 2. Step 3: In (ab), the power of 5 will be (1+2=3).
Correct answer: B. 3
Explanation: Step 1: Powers with the same base are added in multiplication. Step 2: The power of 7 in (a) is 2 and in (b) is 1. Step 3: In (ab), the power of 7 will be (2+1=3).
Correct answer: A. 30
Explanation: Step 1: The factorisation contains 2, 3, and 5. Step 2: (2\times3\times5=30), so the number must be divisible by 30. Step 3: Identify divisibility quickly from prime factors.
Correct answer: A. (3^2\times7^3)
Explanation: Step 1: Write (3087=63\times49). Step 2: (63=3^2\times7) and (49=7^2). Step 3: Therefore, (3087=3^2\times7^3).
Correct answer: C. 8
Explanation: Step 1: In (xy), the powers of the same base 2 are added. Step 2: The power of 2 in (x) is 5 and in (y) is 3. Step 3: So the power of 2 in (xy) is (5+3=8).
Correct answer: A. 360
Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller powers are (2^3), (3^2), and (5^1). Step 3: (8\times9\times5=360), so the HCF is 360.
