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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 72 is (2^3\times3^2). Step 2: Comparing with the given form, the power of 2 is (a). Step 3: Therefore, (a=3).
Correct answer: A. 2
Explanation: Step 1: The prime factorisation of 90 is (2\times3^2\times5). Step 2: In the given form, the power of 3 is (b). Step 3: Comparing gives (b=2).
Correct answer: A. 60
Explanation: Step 1: Calculate (2^2=4). Step 2: (4\times3\times5=60). Step 3: To get the number from prime factorisation, multiply all factors.
Correct answer: A. 72
Explanation: Step 1: Calculate (2^3=8) and (3^2=9). Step 2: (8\times9=72). Step 3: Evaluating powers first gives the answer quickly.
Correct answer: A. 40
Explanation: Step 1: (2^3=8). Step 2: (8\times5=40). Step 3: Multiply the factors to convert prime factorisation into the number.
Correct answer: A. 63
Explanation: The governing concept is evaluation of a numerical expression containing a power and multiplication. In m = 3^2 × 7, the exponent 2 means that 3 is multiplied by itself: 3^2 = 3 × 3 = 9. After evaluating the power, multiply the result by 7: m = 9 × 7 = 63. Therefore, option A is correct. Option B, 42, does not result from correctly evaluating the square. Options C and D also fail because neither equals 9 × 7. The answer can be checked through prime factorisation: 63 = 3 × 3 × 7 = 3^2 × 7, exactly matching the given expression. Thus the numerical value and its factorised form are consistent.
Correct answer: A. (2\times3\times5)
Explanation: Step 1: In prime factorisation, every factor must be prime. Step 2: 2, 3, and 5 are prime. Step 3: 4, 6, and 10 are composite, so they cannot remain in the final prime form.
Correct answer: A. (4\times3\times5)
Explanation: Step 1: A final prime factorisation must not contain a composite number. Step 2: 4 is composite, so (4\times3\times5) is not final form. Step 3: Change 4 into (2^2).
Correct answer: A. (2^6)
Explanation: Step 1: Divide 64 repeatedly by 2. Step 2: Six factors of 2 give (64=2^6). Step 3: 4 and 8 are composite, so do not write them in final prime form.
Correct answer: A. (3^4)
Explanation: Step 1: Divide 81 repeatedly by 3. Step 2: Four factors of 3 give (81=3^4). Step 3: 9 and 27 are composite, so write the power of 3 in final prime form.
Correct answer: A. (11^2)
Explanation: Step 1: 121 is a square number. Step 2: (121=11\times11=11^2). Step 3: Since 11 is prime, (11^2) is the correct prime factorisation.
Correct answer: A. (7^2)
Explanation: Step 1: 49 can be written as (7\times7). Step 2: Since 7 is prime, (49=7^2). Step 3: In a square number, the same prime may appear twice.
Correct answer: A. (3^3)
Explanation: Step 1: Write 27 as (3\times9). Step 2: Since (9=3^2), (27=3^3). Step 3: 9 is composite, so the final form is (3^3).
Correct answer: A. (2^5)
Explanation: Step 1: Divide 32 repeatedly by 2. Step 2: Five factors of 2 give (32=2^5). Step 3: 4 and 8 are composite, so do not write them in final form.
Correct answer: A. (2^4)
Explanation: Step 1: 16 is (2\times2\times2\times2). Step 2: Therefore, (16=2^4). Step 3: 4 and 8 are composite, so write the power of 2 in final prime form.
Correct answer: A. (5^2)
Explanation: Step 1: 25 is (5\times5). Step 2: Since 5 is prime, (25=5^2). Step 3: If the same prime appears twice, write power 2.
Correct answer: A. (2^4\times5)
Explanation: Step 1: Write (80=16\times5). Step 2: Since (16=2^4), (80=2^4\times5). Step 3: 16 is composite, so change it into (2^4).
Correct answer: A. (2\times7^2)
Explanation: Step 1: Write (98=2\times49). Step 2: Since (49=7^2), (98=2\times7^2). Step 3: 49 is composite, so write (7^2) in the final form.
Correct answer: A. (3\times5\times7)
Explanation: Step 1: Write (105=15\times7). Step 2: (15=3\times5), so (105=3\times5\times7). Step 3: 15, 21, and 35 are composite, so do not keep them in the final answer.
Correct answer: A. 36
Explanation: Step 1: Calculate (2^2=4) and (3^2=9). Step 2: (4\times9=36). Step 3: Multiply to get the number from prime factorisation.
Correct answer: A. 50
Explanation: Prime factorisation lists the prime factors with their powers. Compute the power first: \(5^2=25\). Then multiply by 2: \(2\times25=50\). So the number is 50. Why other options are wrong: 25 lacks the factor 2; 100 equals \(2^2\times5^2\) (an extra factor 2); 40 equals \(2^3\times5\). Exam tip: evaluate powers first or multiply the listed prime factors directly to get the number.
Correct answer: A. 42
Explanation: Step 1: The prime factors are given. Step 2: (2\times3\times7=42). Step 3: When no power is written, each prime is taken once.
Correct answer: A. 75
Explanation: Answer: A, 75. Prime factorisation means expressing a number as a product of prime numbers, with powers showing repeated factors. First evaluate the power: 5^2 = 5 × 5 = 25. Then multiply by 3: 3 × 25 = 75. Therefore, the required number is 75. Option A is correct because 75 = 3 × 5 × 5 = 3 × 5^2. Option B, 45, has factorisation 3^2 × 5, so the exponent of 3 is wrong and only one 5 is present. Option C, 90, equals 2 × 3^2 × 5 and contains the extra prime factor 2. Option D, 150, equals 2 × 3 × 5^2 and also contains an unwanted factor 2. Memory cue: calculate powers first, then multiply; do not confuse the exponent with multiplication by the exponent.
Correct answer: A. 48
Explanation: Step 1: Calculate (2^4=16). Step 2: (16\times3=48). Step 3: It is easier to find the value of prime powers first.
Correct answer: A. Composite factor
Explanation: Step 1: A final prime factorisation contains only prime numbers. Step 2: A composite factor must be broken further. Step 3: In exams, do not leave factors like 6, 8, or 10 at the end.
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