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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. (2^2\times3^3)
Explanation: Step 1: Write (108=4\times27). Step 2: (4=2^2) and (27=3^3), so (108=2^2\times3^3). Step 3: 12 and 9 are composite, so they are not the final prime 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, 1, so it is neither prime nor composite. Step 3: Treating 1 as prime is a common mistake.
Correct answer: C. 5
Explanation: Step 1: Look at the given form (96=2^5\times3). Step 2: The exponent written on 2 is 5. Step 3: While reading powers, identify the base and exponent separately.
Correct answer: A. (2\times3^2\times7)
Explanation: Step 1: Write (126=2\times63). Step 2: (63=9\times7=3^2\times7), so (126=2\times3^2\times7). Step 3: 6 and 21 are composite, so they are not final answers.
Correct answer: B. 300
Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (5\times60=300). Step 3: Use this formula when the question is about two numbers.
Correct answer: A. (3^4)
Explanation: Step 1: Write 81 as (9\times9). Step 2: Each 9 is (3^2), so (81=3^4). Step 3: 9 is composite, so (9^2) is not prime factorisation.
Correct answer: A. 2, 3 and 5
Explanation: Step 1: Write (150=15\times10). Step 2: (15=3\times5) and (10=2\times5), so the distinct prime factors are 2, 3, and 5. Step 3: Write a repeated factor only once in a distinct list.
Correct answer: B. 6
Explanation: Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^1) and (3^1), so the HCF is (2\times3=6). Step 3: Take smaller powers for HCF.
Correct answer: C. 72
Explanation: Step 1: For LCM, take higher powers. Step 2: The higher power of (2) is 3 and of (3) is 2, so (2^3\times3^2=72). Step 3: LCM includes the highest powers of all prime factors.
Correct answer: A. (2^3\times5^2)
Explanation: Step 1: Write (200=8\times25). Step 2: (8=2^3) and (25=5^2), so (200=2^3\times5^2). Step 3: 20 and 10 are composite, so they are not prime factorisation.
Correct answer: C. 42
Explanation: Step 1: Multiply the given prime factors. Step 2: (2\times3\times7=42), so the number is 42. Step 3: To get the number from factorisation, multiply all factors.
Correct answer: C. 48
Explanation: Step 1: Find (2^4=16). Step 2: (16\times3=48), so the number is 48. Step 3: First evaluate the power, then multiply.
Correct answer: C. (4\times3)
Explanation: Step 1: In prime factorisation, all factors must be prime. Step 2: In (4\times3), 4 is not prime, so this is not prime factorisation. Step 3: If a composite factor appears, factorise it further.
Correct answer: A. (2^2\times3\times11)
Explanation: Step 1: Write (132=12\times11). Step 2: (12=2^2\times3) and 11 is prime, so (132=2^2\times3\times11). Step 3: Do not leave 12 in the final form because it is composite.
Correct answer: A. They are co-prime
Explanation: Step 1: (14=2\times7) and (25=5^2). Step 2: They have no common prime factor, so they are co-prime. Step 3: Co-prime numbers have HCF 1.
Correct answer: A. (3^2\times5^2)
Explanation: Step 1: Write (225=15\times15). Step 2: Each 15 is (3\times5), so (225=3^2\times5^2). Step 3: 15 is composite, so (15\times15) is not the final prime form.
Correct answer: B. 20
Explanation: Step 1: The common prime factors are 2 and 5. Step 2: The smaller powers are (2^2) and (5^1), so HCF is (4\times5=20). Step 3: For HCF, take smaller powers of common factors only.
Correct answer: C. 120
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: (2^3\times3\times5=8\times3\times5=120). Step 3: In LCM, include prime factors that appear in either number.
Correct answer: B. 36
Explanation: Step 1: A number divisible by both 2 and 3 is also divisible by 6. Step 2: 36 is even and its digit sum is 9, so it is also divisible by 3. Step 3: Divisibility checks help in prime factorisation.
Correct answer: C. 300
Explanation: Step 1: Find (2^2=4) and (5^2=25). Step 2: (4\times3\times25=300). Step 3: In questions with powers, simplify the powers first.
Correct answer: A. Positive integers greater than 1
Explanation: Step 1: This theorem is about writing positive integers greater than 1 as products of primes. Step 2: So it discusses numbers greater than 1. Step 3: Do not include 1 in the usual prime factorisation statement.
Correct answer: A. (2\times3\times5\times7)
Explanation: Step 1: Write (210=21\times10). Step 2: (21=3\times7) and (10=2\times5), so (210=2\times3\times5\times7). Step 3: In the final form, all factors must be prime.
Correct answer: C. 105
Explanation: Step 1: Multiply the given prime factors. Step 2: (3\times5\times7=105), so the number is 105. Step 3: To get the original number from prime factorisation, multiply all factors.
Correct answer: A. (2^2\times3^2\times5)
Explanation: Step 1: Write (180=18\times10). Step 2: (18=2\times3^2) and (10=2\times5), so (180=2^2\times3^2\times5). Step 3: Combine repeated prime factors using powers.
Correct answer: A. Equal to their product
Explanation: Step 1: Co-prime numbers have HCF 1. Step 2: For two numbers, product (=) HCF (\times) LCM. Step 3: Therefore, the LCM of co-prime numbers is equal to their product.
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