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SubjectsMathematics
अभाज्य गुणनखंडन
In Class 10 Mathematics, under the Real Numbers chapter, Prime Factorisation teaches students to express a composite number as a product of prime numbers. Students practise identifying prime factors and writing factorisations using multiplication and exponents. This foundational skill supports the Fundamental Theorem of Arithmetic and later work with HCF and LCM.
Correct answer: A. 762048
Explanation: Step 1: Calculate (2^6=64), (3^5=243), and (7^2=49). Step 2: (64\times243\times49=762048). Step 3: In such calculations, solve powers first.
Correct answer: A. 190512
Explanation: Step 1: Calculate (2^4=16), (3^5=243), and (7^2=49). Step 2: (16\times243\times49=190512). Step 3: Finding all three powers separately is safer.
Correct answer: A. 139392
Explanation: Step 1: Calculate (2^7=128), (3^2=9), and (11^2=121). Step 2: (128\times9\times121=139392). Step 3: Find the values of powers first, then multiply.
Correct answer: A. 99225
Explanation: Step 1: Calculate (3^4=81), (5^2=25), and (7^2=49). Step 2: (81\times25\times49=99225). Step 3: Do not skip square powers in a hurry.
Correct answer: A. 379080
Explanation: Step 1: Calculate (2^3=8) and (3^6=729). Step 2: (8\times729\times5\times13=379080). Step 3: Simplify the higher-power part first.
Correct answer: A. 22
Explanation: Step 1: In a perfect square, all exponents are even. Step 2: The powers of 2 and 11 are odd. Step 3: Multiplying by (2\times11=22) makes both powers even.
Correct answer: A. 21
Explanation: Step 1: For a perfect square, exponents should be even. Step 2: The powers of 3 and 7 are odd. Step 3: Dividing by (3\times7=21) makes the powers 4 and 2.
Correct answer: A. (2\times3^2\times5\times7^2)
Explanation: Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Powers 5, 4, 2, and 7 must become 6, 6, 3, and 9. Step 3: The smallest multiplier is (2\times3^2\times5\times7^2).
Correct answer: A. (2\times3^2\times5^2\times11)
Explanation: Step 1: In a perfect cube, exponents are multiples of 3. Step 2: Reducing 10 to 9, 8 to 6, 5 to 3, and 4 to 3 is the smallest way. Step 3: Therefore, the divisor is (2\times3^2\times5^2\times11).
Correct answer: A. (5^3)
Explanation: Step 1: Every prime power of a divisor must be available in the number. Step 2: (n) has power 2 of 5, but (5^3) needs power 3. Step 3: Therefore, (n) is not divisible by (5^3).
Correct answer: A. (2^8\times3^4\times5^4\times13^2)
Explanation: Step 1: For divisibility, the exponents of the divisor must not exceed those in the number. Step 2: In the first option, all exponents are less than or equal to those in (n). Step 3: Therefore, (n) must be divisible by that number.
Correct answer: A. 23
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (8+6+4+3+2=23). Step 3: Counting only bases and counting with repetition are different.
Correct answer: A. 6
Explanation: Step 1: While counting distinct primes, only bases are counted. Step 2: The bases are 2, 3, 5, 7, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 6.
Correct answer: A. 3
Explanation: Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 11 in (a) is 1 and in (b) is 2. Step 3: In (ab), the power of 11 will be (1+2=3).
Correct answer: A. 9
Explanation: Step 1: Powers with the same base 3 are added in multiplication. Step 2: The power of 3 in (x) is 4 and in (y) is 5. Step 3: The total power will be (4+5=9).
Correct answer: A. 627264
Explanation: Step 1: Calculate (2^6=64), (3^4=81), and (11^2=121). Step 2: (64\times81\times121=627264). Step 3: Solve all three powers separately first.
Correct answer: A. 529200
Explanation: Step 1: Calculate (2^4=16), (3^3=27), (5^2=25), and (7^2=49). Step 2: (16\times27\times25\times49=529200). Step 3: Do the multiplication step by step.
Correct answer: A. (2^5\times3^4\times5^2\times7)
Explanation: Step 1: In the final form, bases must be prime. Step 2: In the first option, bases 2, 3, 5, and 7 are prime. Step 3: 32, 81, 25, 405, 35, and 160 are composite, so they are not final forms.
Correct answer: A. (2^6\times81\times121)
Explanation: Step 1: In an incomplete form, composite bases remain. Step 2: 81 and 121 are composite bases. Step 3: (2^6\times81\times121) must be changed into (2^6\times3^4\times11^2).
Correct answer: A. 680400
Explanation: Step 1: Take (2^4=16), (3^5=243), (5^2=25), and (7). Step 2: (16\times243\times25\times7=680400). Step 3: Solving powers first helps find the correct option quickly.
Correct answer: A. 415800
Explanation: Step 1: Calculate (2^3=8), (3^3=27), and (5^2=25). Step 2: (8\times27\times25\times7\times11=415800). Step 3: When there are many factors, multiply in small groups.
Correct answer: A. (2^4\times3^3\times5^2\times7^2\times11)
Explanation: Step 1: Write the number using prime-base groups. Step 2: (16\times27\times25\times49\times11=2^4\times3^3\times5^2\times7^2\times11). Step 3: Avoid decimal-based options and write prime bases.
Correct answer: A. (2^6\times3^4\times7^2)
Explanation: Step 1: Write (254016=64\times3969). Step 2: (64=2^6) and (3969=3^4\times7^2), so (254016=2^6\times3^4\times7^2). Step 3: Convert 3969 into prime powers.
Correct answer: A. (2^5\times3^5\times5^2\times11)
Explanation: Step 1: Write (2138400=32\times66825). Step 2: (66825=3^5\times5^2\times11), so (2138400=2^5\times3^5\times5^2\times11). Step 3: Give 66825 its complete prime form.
Correct answer: A. Because 64 and 3969 are composite forms
Explanation: Step 1: In final prime factorisation, every base should be prime. Step 2: (64=2^6) and (3969=3^4\times7^2). Step 3: Therefore, the final form is (2^6\times3^4\times7^2).
