Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है
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. 5544
Explanation: Step 1: Calculate (2^3=8) and (3^2=9). Step 2: (8\times9\times7\times11=5544). Step 3: Simplify powers first and then multiply.
Correct answer: A. 6048
Explanation: Step 1: Calculate (2^5=32) and (3^3=27). Step 2: (32\times27\times7=6048). Step 3: Find the values of higher powers separately.
Correct answer: A. 11025
Explanation: Step 1: Calculate (3^2=9), (5^2=25), and (7^2=49). Step 2: (9\times25\times49=11025). Step 3: In this square form, all exponents are 2, so multiply carefully.
Correct answer: A. 11760
Explanation: Step 1: Calculate (2^4=16) and (7^2=49). Step 2: (16\times3\times5\times49=11760). Step 3: Calculating powers first makes the work easier.
Correct answer: A. 31104
Explanation: Step 1: Calculate (2^7=128) and (3^5=243). Step 2: (128\times243=31104). Step 3: Simplifying both powers first is the correct method.
Correct answer: A. 14
Explanation: Step 1: For a perfect square, all exponents must be even. Step 2: The powers of 2 and 7 are odd. Step 3: Multiplying by (2\times7=14) makes all exponents even.
Correct answer: A. 3
Explanation: Step 1: In a perfect square, all exponents are even. Step 2: Only the power of 3 is odd. Step 3: Dividing by 3 makes the power 4 and the number becomes a perfect square.
Correct answer: A. 588
Explanation: Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Powers 4, 2, and 1 must become 6, 3, and 3 respectively. Step 3: The multiplier is (2^2\times3\times7^2=588).
Correct answer: A. (2^2\times3^2\times5\times11^2)
Explanation: Step 1: For a perfect cube, exponents should be multiples of 3. Step 2: Reduce 8 to 6, 5 to 3, 4 to 3, and 2 to 0 for the smallest divisor. Step 3: So the divisor is (2^2\times3^2\times5\times11^2).
Correct answer: A. (3^5)
Explanation: Step 1: For divisibility, every prime power of the divisor must be available in the number. Step 2: (n) has power 4 of 3, but (3^5) needs power 5. Step 3: Therefore, (n) is not divisible by (3^5).
Correct answer: A. (2^7\times3^2\times5^4)
Explanation: Step 1: For divisibility, each exponent in the divisor must be less than or equal to the corresponding exponent in the number. Step 2: (2^7), (3^2), and (5^4) are all available in (n). Step 3: Therefore, (n) must be divisible by (2^7\times3^2\times5^4).
Correct answer: A. 10
Explanation: Step 1: To count with repetition, add the exponents. Step 2: (4+2+2+2=10). Step 3: Keep the difference between distinct prime count and repeated prime count clear.
Correct answer: A. 4
Explanation: Step 1: While counting distinct prime factors, do not add exponents. Step 2: The prime bases are 2, 3, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 4.
Correct answer: A. 6
Explanation: Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 3 in (a) is 2 and in (b) is 4. Step 3: In (ab), the power of 3 will be (2+4=6).
Correct answer: A. 8
Explanation: Step 1: Powers with the same base 5 are added in multiplication. Step 2: The power of 5 in (x) is 3 and in (y) is 5. Step 3: The total power will be (3+5=8).
Correct answer: A. 21168
Explanation: Step 1: Calculate (2^4=16), (3^3=27), and (7^2=49). Step 2: (16\times27\times49=21168). Step 3: Solve all three powers separately first.
Correct answer: A. 28224
Explanation: Step 1: Calculate (2^6=64), (3^2=9), and (7^2=49). Step 2: (64\times9\times49=28224). Step 3: Do multiplication step by step to avoid mistakes.
Correct answer: A. (2^4\times3^2\times5\times7^2)
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: 16, 45, 49, and 4410 are composite, so they are not final forms.
Correct answer: A. (2^5\times21^2)
Explanation: Step 1: In an incomplete form, a composite base remains. Step 2: 21 is composite and (21=3\times7), so (2^5\times21^2) is not final. Step 3: Change it into (2^5\times3^2\times7^2).
Correct answer: A. 28224
Explanation: Step 1: (2^6=64), (3^2=9), and (7^2=49). Step 2: (64\times9\times49=28224). Step 3: Solving powers first helps find the correct option quickly.
Correct answer: A. 19600
Explanation: Step 1: Calculate (2^4=16), (5^2=25), and (7^2=49). Step 2: (16\times25\times49=19600). Step 3: Finding all three powers separately is safer.
Correct answer: A. (2^5\times5^2\times7^2)
Explanation: Step 1: Write (39200=32\times1225). Step 2: (32=2^5) and (1225=5^2\times7^2), so (39200=2^5\times5^2\times7^2). Step 3: Convert 1225 into prime powers.
Correct answer: A. (2^2\times3^3\times5\times7\times11)
Explanation: Step 1: Write (41580=4\times10395). Step 2: (10395=3^3\times5\times7\times11), so (41580=2^2\times3^3\times5\times7\times11). Step 3: Give 10395 its complete prime form.
Correct answer: A. (2^8\times3^5)
Explanation: Step 1: Write (62208=256\times243). Step 2: (256=2^8) and (243=3^5), so (62208=2^8\times3^5). Step 3: Write 256 and 243 as prime powers.
Correct answer: A. Because 8 and 3465 are composite forms
Explanation: Step 1: In prime factorisation, every final factor should be in prime-base form. Step 2: (8=2^3) and (3465=3^2\times5\times7\times11). Step 3: Therefore, the final form is (2^3\times3^2\times5\times7\times11).
