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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. (3^2\times5^2\times7^2)
Explanation: Step 1: Recognise (11025=105^2). Step 2: Since (105=3\times5\times7), (105^2=3^2\times5^2\times7^2). Step 3: In perfect squares, every prime exponent is even.
Correct answer: A. 315
Explanation: Step 1: The prime factorisation of 315 is (3^2\times5\times7). Step 2: All these factors are present in the prime factorisation of (n). Step 3: Therefore, (n) must be divisible by 315.
Correct answer: C. 48
Explanation: Step 1: In (n), the powers are (2^4), (3^1), and (5^2). Step 2: (48=2^4\times3), so divisibility by 48 is possible. Step 3: (80=2^4\times5), (75=3\times5^2), and (150=2\times3\times5^2) are also present; hence all given options divide (n).
Correct answer: A. 2880
Explanation: Step 1: Calculate (2^6=64) and (3^2=9). Step 2: (64\times9\times5=2880). Step 3: In larger products, evaluate powers first and then multiply.
Correct answer: A. 16200
Explanation: Step 1: Calculate (2^3=8), (3^4=81), and (5^2=25). Step 2: (8\times81\times25=16200). Step 3: Simplifying powers first keeps the calculation safe.
Correct answer: C. 4
Explanation: Step 1: In LCM, take the higher power of each prime. Step 2: The powers of 2 are 2 and 4. Step 3: The higher power is 4, so the answer is 4.
Correct answer: B. 1
Explanation: Step 1: In HCF, take the smaller power of the common prime. Step 2: The powers of 5 are 1 and 2. Step 3: The smaller power is 1, so the answer is 1.
Correct answer: A. (3^4\times7^2)
Explanation: Step 1: (3969=63^2). Step 2: Since (63=3^2\times7), (63^2=3^4\times7^2). Step 3: (63\times63) gives the product, but it is not prime factorisation.
Correct answer: B. (2^2\times3\times5)
Explanation: Step 1: For a perfect cube, exponents must be multiples of 3. Step 2: We need (2), (3), and (5^2) to make the powers (6,3,6). Step 3: The smallest multiplier is (2\times3\times5^2).
Correct answer: C. 6
Explanation: Step 1: (3528=72\times49=2^3\times3^2\times7^2). Step 2: For a perfect square, all exponents must be even, but the exponent of 2 is 3. Step 3: Dividing by 2 gives (2^2\times3^2\times7^2), so the smallest number is 2.
Correct answer: C. 5
Explanation: Step 1: In multiplication, add the exponents of the same prime base. Step 2: The power of 3 in (x) is 2 and in (y) is 3. Step 3: In (xy), the power of 3 is (2+3=5).
Correct answer: C. 8
Explanation: Step 1: Powers of the same base 2 are added in multiplication. Step 2: The power of 2 in (x) is 5 and in (y) is 3. Step 3: The total power is (5+3=8).
Correct answer: A. 2, 3, 5, 7 and 11
Explanation: Step 1: Write (2310=231\times10). Step 2: (231=3\times7\times11) and (10=2\times5). Step 3: The distinct prime factors are 2, 3, 5, 7, and 11.
Correct answer: A. 132
Explanation: Step 1: The common prime factors are 2, 3, and 11. Step 2: The smaller powers are (2^2), (3^1), and (11^1). Step 3: (4\times3\times11=132), so the HCF is 132.
Correct answer: A. 43560
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^3), (3^2), (5), and (11^2). Step 3: (8\times9\times5\times121=43560), so the answer is 43560.
Correct answer: C. 4
Explanation: Step 1: The product is (36\times900). Step 2: (36=2^2\times3^2) and (900=2^2\times3^2\times5^2). Step 3: The power of 2 in the product is (2+2=4).
Correct answer: B. 1
Explanation: Step 1: The product is (30\times2310). Step 2: 30 has no factor 11 and (2310=2\times3\times5\times7\times11). Step 3: Therefore, the power of 11 in the product is (0+1=1).
Correct answer: A. (7^4)
Explanation: Step 1: (2401=49\times49). Step 2: Since (49=7^2), (2401=7^4). Step 3: 49 is composite, so (49^2) is not the final prime factorisation.
Correct answer: A. (2^2\times3^2)
Explanation: Step 1: In a perfect cube, each exponent must be a multiple of 3. Step 2: (2^2) and (3^5) cause the issue; reduce them to 0 and 3. Step 3: Dividing by (2^2\times3^2) gives exponents (0,3,3).
Correct answer: A. 2520
Explanation: Step 1: First calculate (2^3=8) and (3^2=9). Step 2: (8\times9\times5\times7=2520). Step 3: To get the number from prime factorisation, multiply all factors.
Correct answer: A. (ab)
Explanation: Step 1: For two numbers, HCF (\times) LCM equals the product of the two numbers. Step 2: This also follows from prime powers because the smaller and higher exponents together give the total exponent. Step 3: Therefore, the answer is (ab).
Correct answer: A. 12
Explanation: Step 1: HCF is (2^3\times3=24). Step 2: LCM is (2^5\times3^2=288). Step 3: The ratio is (288\div24=12).
Correct answer: D. 13
Explanation: Step 1: In (xy), exponents of the same bases are added. Step 2: Powers become (2^{6}), (3^{4}), and (5^{3}). Step 3: Counting with repetition gives (6+4+3=13).
Correct answer: C. 3
Explanation: Step 1: In LCM, take the higher power. Step 2: The powers of 7 are 1 and 3. Step 3: The higher power is 3, so the answer is 3.
Correct answer: B. 2
Explanation: Step 1: In HCF, take the smaller power. Step 2: The powers of 3 are 2 and 4. Step 3: The smaller power is 2, so the answer is 2.
