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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. 3
Explanation: Step 1: The prime factorisation of 600 is (2^3\times3\times5^2). Step 2: In the given form, the power of 2 is (a). Step 3: Comparing gives (a=3).
Correct answer: A. 3
Explanation: Step 1: (675=27\times25). Step 2: (27=3^3) and (25=5^2), so (675=3^3\times5^2). Step 3: Comparing gives (a=3).
Correct answer: A. 720
Explanation: Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5=720). Step 3: Evaluating powers first makes calculation easier.
Correct answer: A. 162
Explanation: This question applies the meaning of an exponent in a prime factorisation. In 2 × 3^4, calculate the power first: 3^4 = 3 × 3 × 3 × 3 = 81. Then multiply by the factor 2: 2 × 81 = 162. Therefore option A, 162, is correct. Option B is only 3^4 and omits the factor 2. Option C is twice the correct result, and option D includes an incorrect extra factor. A factorisation check confirms the result: 162 = 2 × 81 = 2 × 3^4. This illustrates that a prime factorisation records both the prime factors and their multiplicities, represented here by the exponent 4 on 3.
Correct answer: A. 168
Explanation: Step 1: (2^3=8). Step 2: (8\times3\times7=168). Step 3: Multiply to get the number from prime factorisation.
Correct answer: A. 225
Explanation: The governing idea is evaluating each power in a factorised numerical expression and then multiplying the results. Here, 3^2 = 3 × 3 = 9 and 5^2 = 5 × 5 = 25. Therefore m = 9 × 25 = 225. Option A is correct. Another efficient check is to group equal exponents: 3^2 × 5^2 = (3 × 5)^2 = 15^2 = 225, using the law a^n b^n = (ab)^n. Option B is too small and reflects an incorrect calculation, while 300 and 450 do not follow from the stated factors. The prime factorisation of 225 is 3^2 × 5^2, confirming the answer independently.
Correct answer: A. (2\times3\times5\times7\times11)
Explanation: Step 1: In final prime factorisation, every factor must be prime. Step 2: 2, 3, 5, 7, and 11 are prime. Step 3: 6, 21, 35, and 33 are composite, so they cannot remain in final form.
Correct answer: A. (10\times3^2\times7)
Explanation: Step 1: Final prime factorisation must not contain a composite factor. Step 2: 10 is composite, so (10\times3^2\times7) is not final form. Step 3: Change 10 into (2\times5).
Correct answer: A. (2^{11})
Explanation: Step 1: Divide 2048 repeatedly by 2. Step 2: Eleven factors of 2 give (2048=2^{11}). Step 3: 4, 32, and 64 are composite, so write the power of 2 in final form.
Correct answer: A. (3^6)
Explanation: Step 1: Divide 729 repeatedly by 3. Step 2: Six factors of 3 give (729=3^6). Step 3: 9 and 27 are composite, so keep prime base 3.
Correct answer: A. (19^2)
Explanation: Step 1: (361=19\times19). Step 2: Since 19 is prime, (361=19^2). Step 3: In a square number, the same prime appears twice.
Correct answer: A. (11^3)
Explanation: Step 1: Write (1331=11\times121). Step 2: (121=11^2), so (1331=11^3). Step 3: 121 is composite, so write (11^3) in the final form.
Correct answer: A. (2^{12})
Explanation: Step 1: Divide 4096 repeatedly by 2. Step 2: Twelve factors of 2 give (4096=2^{12}). Step 3: 64 and 16 are composite, so they are not final prime forms.
Correct answer: A. (5^5)
Explanation: Step 1: Write (3125=5\times625). Step 2: Since (625=5^4), (3125=5^5). Step 3: 25 and 125 are composite forms, so write a power of 5 in the final answer.
Correct answer: A. (2^3\times7^3)
Explanation: Step 1: Recognise (2744=14^3). Step 2: Since (14=2\times7), (2744=2^3\times7^3). Step 3: 14 is composite, so write prime bases in the final form.
Correct answer: A. (2^9\times3)
Explanation: Step 1: Write (1536=512\times3). Step 2: (512=2^9), so (1536=2^9\times3). Step 3: Do not keep 512 in the final form; write (2^9).
Correct answer: A. (2^3\times3^3\times5)
Explanation: Step 1: Write (1080=108\times10). Step 2: (108=2^2\times3^3) and (10=2\times5), so (1080=2^3\times3^3\times5). Step 3: Count powers of 2 and 3 separately.
Correct answer: A. (2\times3\times5\times7^2)
Explanation: Step 1: Write (1470=30\times49). Step 2: (30=2\times3\times5) and (49=7^2), so (1470=2\times3\times5\times7^2). Step 3: Convert 49 into (7^2).
Correct answer: A. (2^2\times3^2\times5\times11)
Explanation: Step 1: Write (1980=198\times10). Step 2: (198=2\times3^2\times11) and (10=2\times5), so (1980=2^2\times3^2\times5\times11). Step 3: Since 2 appears twice, write (2^2).
Correct answer: A. 1176
Explanation: Step 1: Calculate (2^3=8) and (7^2=49). Step 2: (8\times3\times49=1176). Step 3: Evaluating powers first makes calculation easier.
Correct answer: A. 1225
Explanation: Step 1: (5^2=25) and (7^2=49). Step 2: (25\times49=1225). Step 3: Both powers are 2 in this square form, so multiply carefully.
Correct answer: A. 2310
Explanation: Step 1: Multiply all given prime factors. Step 2: (2\times3\times5\times7\times11=2310). Step 3: When no power is written, each prime is taken once.
Correct answer: A. 1800
Explanation: Step 1: Calculate (2^3=8), (3^2=9), and (5^2=25). Step 2: (8\times9\times25=1800). Step 3: Simplify powers first, then multiply.
Correct answer: A. 1296
Explanation: Step 1: Calculate (2^4=16) and (3^4=81). Step 2: (16\times81=1296). Step 3: Evaluate prime powers and then multiply.
Correct answer: A. Writing repeated prime factors shortly and clearly
Explanation: Step 1: When a prime repeats, powers are used. Step 2: For example, (2\times2\times2) is written as (2^3). Step 3: This makes the answer short, clear, and easy to read in exams.
