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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. 2376
Explanation: Step 1: Calculate (2^3=8) and (3^3=27). Step 2: (8\times27\times11=2376). Step 3: Finding powers first is the correct method.
Correct answer: A. 3465
Explanation: Step 1: Calculate (3^2=9). Step 2: (9\times5\times7\times11=3465). Step 3: When there are many factors, multiply in pairs.
Correct answer: A. 5040
Explanation: Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5\times7=5040). Step 3: First do (16\times9=144), then multiply the rest.
Correct answer: A. 5292
Explanation: Step 1: Calculate (2^2=4), (3^3=27), and (7^2=49). Step 2: (4\times27\times49=5292). Step 3: Simplify all three powers separately.
Correct answer: A. 10368
Explanation: Step 1: Calculate (2^7=128) and (3^4=81). Step 2: (128\times81=10368). Step 3: Solving large powers first reduces mistakes.
Correct answer: A. 110
Explanation: Step 1: In a perfect square, all exponents are even. Step 2: Powers of 2, 5, and 11 are odd. Step 3: Multiplying by (2\times5\times11=110) makes all exponents even.
Correct answer: A. 3
Explanation: Step 1: In a perfect square, all exponents should be even. Step 2: Only the power of 3 is odd. Step 3: Dividing by 3 makes the power of 3 equal to 4 and the number becomes a perfect square.
Correct answer: A. (2\times3^2\times5\times7^2)
Explanation: Step 1: In a perfect cube, exponents are multiples of 3. Step 2: Powers 5, 4, 2, and 1 must become 6, 6, 3, and 3. Step 3: The smallest multiplier is (2\times3^2\times5\times7^2).
Correct answer: A. (2^2\times3\times5)
Explanation: Step 1: For a perfect cube, exponents should be multiples of 3. Step 2: Reducing 8 to 6, 7 to 6, and 4 to 3 is the smallest way. Step 3: Therefore, the divisor is (2^2\times3\times5).
Correct answer: A. (2^6)
Explanation: Step 1: Every prime power of a divisor must be available in the number. Step 2: (n) has power 5 of 2, but (2^6) needs 6. Step 3: Therefore, (n) is not divisible by (2^6).
Correct answer: A. (2^5\times3\times5^4\times11)
Explanation: Step 1: For divisibility, exponents in the divisor must not exceed those in the number. Step 2: (2^5), (3), (5^4), and 11 are all available in (n). Step 3: Therefore, (n) must be divisible by this number.
Correct answer: A. 15
Explanation: Step 1: For counting with repetition, add the exponents. Step 2: (6+4+3+2=15). Step 3: Remember the difference between the number of distinct bases and the count with repetition.
Correct answer: A. 4
Explanation: Step 1: While counting distinct primes, only bases are counted. Step 2: The bases are 2, 3, 11, and 13. Step 3: Therefore, there are 4 distinct prime factors.
Correct answer: A. 3
Explanation: Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 7 in (a) is 1 and in (b) is 2. Step 3: In (ab), the power of 7 will be (1+2=3).
Correct answer: A. 3
Explanation: Step 1: Powers of the same base 11 are added in multiplication. Step 2: The power of 11 in (x) is 2 and in (y) is 1. Step 3: The total power will be (2+1=3).
Correct answer: A. 35280
Explanation: Step 1: Calculate (2^4=16), (3^2=9), and (7^2=49). Step 2: (16\times9\times5\times49=35280). Step 3: Solve powers separately first.
Correct answer: A. 20736
Explanation: Step 1: Calculate (2^8=256) and (3^4=81). Step 2: (256\times81=20736). Step 3: Simplify higher powers separately and multiply.
Correct answer: A. (2^3\times3^2\times5\times7\times11)
Explanation: Step 1: In the final form, bases must be prime. Step 2: In the first form, the bases 2, 3, 5, 7, and 11 are prime. Step 3: 8, 9, 385, and 72 are composite, so they are not final forms.
Correct answer: A. (2^4\times45\times49)
Explanation: Step 1: In an incomplete form, composite factors remain. Step 2: Both 45 and 49 are composite. Step 3: (2^4\times45\times49) must be changed into (2^4\times3^2\times5\times7^2).
Correct answer: A. 5040
Explanation: Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5\times7=5040). Step 3: Solving powers first helps find the correct option quickly.
Correct answer: A. 9800
Explanation: Step 1: Calculate (2^3=8), (5^2=25), and (7^2=49). Step 2: (8\times25\times49=9800). Step 3: Find all three powers separately.
Correct answer: A. (2^3\times3\times5^2\times7^2)
Explanation: Step 1: Write (29400=600\times49). Step 2: (600=2^3\times3\times5^2) and (49=7^2), so (29400=2^3\times3\times5^2\times7^2). Step 3: Break 600 and 49 completely.
Correct answer: A. (3^3\times5^2\times7^2)
Explanation: Step 1: Write (33075=675\times49). Step 2: (675=3^3\times5^2) and (49=7^2), so (33075=3^3\times5^2\times7^2). Step 3: Write 675 and 49 as prime powers.
Correct answer: A. (2^6\times3^6)
Explanation: Step 1: (46656) can be written as (64\times729). Step 2: (64=2^6) and (729=3^6), so (46656=2^6\times3^6). Step 3: Do not leave 64 and 729 in the final form.
Correct answer: A. Because 600 and 49 are composite forms
Explanation: Step 1: In final prime factorisation, every base should be prime. Step 2: (600=2^3\times3\times5^2) and (49=7^2). Step 3: Therefore, the final form is (2^3\times3\times5^2\times7^2).
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