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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. 14256
Explanation: Step 1: Calculate (2^4=16) and (3^4=81). Step 2: (16\times81\times11=14256). Step 3: Solve powers first, then multiply by 11.
Correct answer: A. 18144
Explanation: Step 1: Calculate (2^5=32) and (3^4=81). Step 2: (32\times81\times7=18144). Step 3: It is better to find higher powers first.
Correct answer: A. 25410
Explanation: Step 1: Calculate (11^2=121). Step 2: (2\times3\times5\times7\times121=25410). Step 3: First take the product of smaller factors as 210, then multiply by 121.
Correct answer: A. 43200
Explanation: Step 1: Calculate (2^6=64), (3^3=27), and (5^2=25). Step 2: (64\times27\times25=43200). Step 3: Simplify all three powers separately.
Correct answer: A. 127008
Explanation: Step 1: Calculate (2^5=32), (3^4=81), and (7^2=49). Step 2: (32\times81\times49=127008). Step 3: Solving powers first keeps multiplication clear.
Correct answer: A. 21
Explanation: Step 1: In a perfect square, all exponents should be even. Step 2: The powers of 3 and 7 are odd. Step 3: Multiplying by (3\times7=21) makes both powers even.
Correct answer: A. (2\times5)
Explanation: Step 1: For a perfect square, exponents should be even. Step 2: The powers of 2 and 5 are odd. Step 3: Dividing by (2\times5) makes the powers 8 and 6.
Correct answer: A. (2\times3^2\times5^2\times13)
Explanation: Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Powers 5, 7, 4, and 2 must become 6, 9, 6, and 3. Step 3: The smallest multiplier is (2\times3^2\times5^2\times13).
Correct answer: A. (2\times3^2\times5^2\times7)
Explanation: Step 1: For a perfect cube, exponents should be 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\times7).
Correct answer: A. (11^3)
Explanation: Step 1: Every prime power of a divisor must be available in the number. Step 2: (n) has power 2 of 11, but (11^3) needs power 3. Step 3: Therefore, (n) is not divisible by (11^3).
Correct answer: A. (2^9\times3^4\times7^3\times13)
Explanation: Step 1: For divisibility, exponents in the divisor must not exceed those in the given number. Step 2: (2^9), (3^4), (7^3), and 13 are all available in (n). Step 3: Therefore, (n) must be divisible by the first option.
Correct answer: A. 25
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (9+7+4+3+2=25). Step 3: Keep the number of bases and the total count with repetition separate.
Correct answer: A. 5
Explanation: Step 1: While counting distinct primes, only bases are counted. Step 2: The bases are 2, 3, 7, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 5.
Correct answer: A. 9
Explanation: Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 3 in (a) is 4 and in (b) is 5. Step 3: In (ab), the power of 3 will be (4+5=9).
Correct answer: A. 4
Explanation: Step 1: Powers with the same base 13 are added in multiplication. Step 2: The power of 13 in (x) is 1 and in (y) is 3. Step 3: The total power will be (1+3=4).
Correct answer: A. 127008
Explanation: Step 1: Calculate (2^5=32), (3^4=81), and (7^2=49). Step 2: (32\times81\times49=127008). Step 3: Solve all three powers separately first.
Correct answer: A. 95256
Explanation: Step 1: Calculate (2^3=8), (3^5=243), and (7^2=49). Step 2: (8\times243\times49=95256). Step 3: Simplify higher powers first.
Correct answer: A. (2^3\times3^3\times5\times7^2)
Explanation: Step 1: In the final form, every base must be prime. Step 2: In the first option, bases 2, 3, 5, and 7 are prime. Step 3: 8, 135, 49, 27, 245, and 216 are composite, so they are not final forms.
Correct answer: A. (2^5\times81\times49)
Explanation: Step 1: In an incomplete form, composite bases remain. Step 2: 81 and 49 are composite bases. Step 3: (2^5\times81\times49) must be changed into (2^5\times3^4\times7^2).
Correct answer: A. 43200
Explanation: Step 1: (2^6=64), (3^3=27), and (5^2=25). Step 2: (64\times27\times25=43200). Step 3: Solving powers first helps find the correct option quickly.
Correct answer: A. 108900
Explanation: Step 1: Calculate (2^2=4), (3^2=9), (5^2=25), and (11^2=121). Step 2: (4\times9\times25\times121=108900). Step 3: This is a square form, so observe the powers carefully.
Correct answer: A. (2^3\times3^4\times5\times7^2)
Explanation: Step 1: Write (158760=8\times19845). Step 2: (19845=3^4\times5\times7^2), so (158760=2^3\times3^4\times5\times7^2). Step 3: Break 19845 completely.
Correct answer: A. (2^3\times3^2\times5^2\times11^2)
Explanation: Step 1: Write (217800=8\times27225). Step 2: (27225=3^2\times5^2\times11^2), so (217800=2^3\times3^2\times5^2\times11^2). Step 3: Convert 27225 into prime powers.
Correct answer: A. (2^7\times3^7)
Explanation: Step 1: (279936) can be written as (128\times2187). Step 2: (128=2^7) and (2187=3^7), so (279936=2^7\times3^7). Step 3: Do not leave 128 and 2187 in the final form.
Correct answer: A. Because 8 and 27225 are composite forms
Explanation: Step 1: In final prime factorisation, every base should be prime. Step 2: (8=2^3) and (27225=3^2\times5^2\times11^2). Step 3: Therefore, the final form is (2^3\times3^2\times5^2\times11^2).
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