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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\times5^2\times7^2)
Explanation: Step 1: Write (14700=147\times100). Step 2: (147=3\times7^2) and (100=2^2\times5^2), so (14700=2^2\times3\times5^2\times7^2). Step 3: 147 and 100 are composite, so do not keep them in the final form.
Correct answer: A. 1485
Explanation: Step 1: (1485=3^3\times5\times11). Step 2: All these factors are present in the prime factorisation of (n). Step 3: Therefore, (n) must be divisible by 1485.
Correct answer: D. 75
Explanation: Step 1: In (n), the powers are (2^4), (3^2), and (5^1). Step 2: (75=3\times5^2), which needs power 2 of 5. Step 3: Since (n) has only (5^1), (n) will not be divisible by 75.
Correct answer: A. 5760
Explanation: Step 1: Calculate (2^7=128) and (3^2=9). Step 2: (128\times9\times5=5760). Step 3: In larger products, simplifying powers first is safer.
Correct answer: A. 97200
Explanation: Step 1: Calculate (2^4=16), (3^5=243), and (5^2=25). Step 2: (16\times243\times25=97200). Step 3: Simplifying powers first makes the calculation easier.
Correct answer: C. 6
Explanation: Step 1: In LCM, take the higher power of each prime. Step 2: The powers of 2 are 3 and 6. Step 3: The higher power is 6, so the answer is 6.
Correct answer: A. 2
Explanation: Step 1: In HCF, take the smaller power of the common prime. Step 2: The powers of 3 are 5 and 2. Step 3: The smaller power is 2, so the answer is 2.
Correct answer: A. (2^2\times3^4\times7^2)
Explanation: Step 1: (15876=126^2). Step 2: Since (126=2\times3^2\times7), (126^2=2^2\times3^4\times7^2). Step 3: In a perfect square, every prime exponent is even.
Correct answer: A. (2^2\times3\times5^2)
Explanation: Step 1: For a perfect cube, exponents must be multiples 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. 15
Explanation: Step 1: (6615=3^3\times5\times7^2). Step 2: For a perfect square, all exponents must be even, but powers of 3 and 5 are odd. Step 3: Dividing by (3\times5=15) leaves (3^2\times7^2), a perfect square.
Correct answer: C. 7
Explanation: Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 3 in (x) is 3 and in (y) is 4. Step 3: In (xy), the power of 3 is (3+4=7).
Correct answer: C. 10
Explanation: Step 1: Powers of the same base 2 are added in multiplication. Step 2: The power of 2 in (x) is 6 and in (y) is 4. Step 3: The total power is (6+4=10).
Correct answer: A. 2, 3, 5, 7, 11 and 13
Explanation: Step 1: Write (30030=2310\times13). Step 2: (2310=2\times3\times5\times7\times11). Step 3: Therefore, the distinct prime factors are 2, 3, 5, 7, 11, and 13.
Correct answer: A. 156
Explanation: Step 1: The common prime factors are 2, 3, and 13. Step 2: The smaller powers are (2^2), (3^1), and (13^1). Step 3: (4\times3\times13=156), so the HCF is 156.
Correct answer: A. 365040
Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^4), (3^3), (5), and (13^2). Step 3: (16\times27\times5\times169=365040), so the answer is 365040.
Correct answer: B. 4
Explanation: Step 1: The product is (72\times1800). Step 2: (72=2^3\times3^2) and (1800=2^3\times3^2\times5^2). Step 3: The power of 3 in the product is (2+2=4).
Correct answer: B. 1
Explanation: Step 1: The product is (42\times30030). Step 2: 42 has no factor 13 and (30030=2\times3\times5\times7\times11\times13). Step 3: Therefore, the power of 13 in the product is (0+1=1).
Correct answer: A. (11^4)
Explanation: Step 1: (14641=121\times121). Step 2: Since (121=11^2), (14641=11^4). Step 3: 121 is composite, so (121^2) is not the final prime factorisation.
Correct answer: A. (2^2\times3)
Explanation: Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Reduce (2^5) to (2^3) by dividing by (2^2), and reduce (3^4) to (3^3) by dividing by 3. Step 3: So the smallest divisor is (2^2\times3).
Correct answer: A. 15120
Explanation: Step 1: First calculate (2^4=16) and (3^3=27). Step 2: (16\times27\times5\times7=15120). 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: In prime powers, the smaller and higher exponents together give the total exponent. Step 3: Therefore, the answer is (ab).
Correct answer: A. 72
Explanation: Step 1: HCF is (2^4\times3=48). Step 2: LCM is (2^7\times3^3=3456). Step 3: The ratio is (3456\div48=72).
Correct answer: C. 19
Explanation: Step 1: In (xy), exponents of the same bases are added. Step 2: Powers become (2^8), (3^6), and (5^5). Step 3: Counting with repetition gives (8+6+5=19).
Correct answer: C. 4
Explanation: Step 1: In LCM, take the higher power. Step 2: The powers of 7 are 2 and 4. Step 3: The higher power is 4, so the answer is 4.
Correct answer: B. 3
Explanation: Step 1: In HCF, take the smaller power. Step 2: The powers of 3 are 3 and 5. Step 3: The smaller power is 3, so the answer is 3.
