Which number has prime factorisation (2^6\times3^5\times7^2)?
Correct answer: A
Step 1: Calculate (2^6=64), (3^5=243), and (7^2=49). Step 2: (64\times243\times49=762048). Step 3: In such calculations, solve powers first.
Mathematics
Prime Factorisation
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.
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Expert · Level 4
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- 762048
- 381024
- 1524096
- 190512
Expert · Level 4
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- 190512
- 95256
- 381024
- 127008
Expert · Level 4
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- 139392
- 69696
- 278784
- 177408
Expert · Level 4
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- 99225
- 49612
- 198450
- 11025
Expert · Level 4
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- 379080
- 189540
- 758160
- 117000
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- 22
- 2
- 11
- 44
Expert · Level 4
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- 21
- 3
- 7
- 63
Expert · Level 4
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- (2\times3^2\times5\times7^2)
- (2\times3\times5\times7)
- (3^2\times7^2)
- (2^2\times5\times7)
Expert · Level 4
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- (2\times3^2\times5^2\times11)
- (2^2\times3\times5\times11)
- (2\times3\times5^2\times11^2)
- (3^2\times5\times11)
Expert · Level 4
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- (5^3)
- (2^7\times3^5)
- (3^6\times7^3)
- (2^8\times5^2)
Expert · Level 4
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- (2^8\times3^4\times5^4\times13^2)
- (2^{10}\times3^3)
- (3^5\times5^2)
- (2^9\times13^3)
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- 23
- 5
- 20
- 25
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- 6
- 21
- 5
- 7
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- 3
- 2
- 1
- 4
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- 9
- 5
- 4
- 8
Expert · Level 4
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- 627264
- 313632
- 1254528
- 139392
Expert · Level 4
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- 529200
- 264600
- 1058400
- 415800
Expert · Level 4
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- (2^5\times3^4\times5^2\times7)
- (32\times81\times25\times7)
- (2^5\times405\times35)
- (160\times2835)
Expert · Level 4
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- (2^6\times81\times121)
- (2^6\times3^4\times11^2)
- (2^3\times3^6\times5\times13)
- (3^4\times5^2\times7^2)
Expert · Level 4
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- 680400
- 340200
- 1360800
- 453600
Expert · Level 4
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- 415800
- 207900
- 831600
- 277200
Expert · Level 4
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- (2^4\times3^3\times5^2\times7^2\times11)
- (2^5\times3^2\times5^2\times7^2\times11)
- (16\times36382.5)
- (2^4\times27\times25\times49\times11)
Expert · Level 4
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- (2^6\times3^4\times7^2)
- (2^5\times3^5\times7^2)
- (64\times3969)
- (2^6\times81\times49)
Expert · Level 4
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- (2^5\times3^5\times5^2\times11)
- (2^6\times3^4\times5^2\times11)
- (32\times66825)
- (2^5\times243\times25\times11)
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- Because 64 and 3969 are composite forms
- Because 64 is prime
- Because 3969 cannot be factorised
- Because the product will change
Which number has prime factorisation (2^4\times3^5\times7^2)?
Correct answer: A
Step 1: Calculate (2^4=16), (3^5=243), and (7^2=49). Step 2: (16\times243\times49=190512). Step 3: Finding all three powers separately is safer.
Which number has prime factorisation (2^7\times3^2\times11^2)?
Correct answer: A
Step 1: Calculate (2^7=128), (3^2=9), and (11^2=121). Step 2: (128\times9\times121=139392). Step 3: Find the values of powers first, then multiply.
Which number has prime factorisation (3^4\times5^2\times7^2)?
Correct answer: A
Step 1: Calculate (3^4=81), (5^2=25), and (7^2=49). Step 2: (81\times25\times49=99225). Step 3: Do not skip square powers in a hurry.
Which number has prime factorisation (2^3\times3^6\times5\times13)?
Correct answer: A
Step 1: Calculate (2^3=8) and (3^6=729). Step 2: (8\times729\times5\times13=379080). Step 3: Simplify the higher-power part first.
If the number is (2^7\times3^4\times5^2\times11^3), by which smallest number should it be multiplied to make a perfect square?
Correct answer: A
Step 1: In a perfect square, all exponents are even. Step 2: The powers of 2 and 11 are odd. Step 3: Multiplying by (2\times11=22) makes both powers even.
If the number is (2^8\times3^5\times5^4\times7^3), by which smallest number should it be divided to make a perfect square?
Correct answer: A
Step 1: For a perfect square, exponents should be even. Step 2: The powers of 3 and 7 are odd. Step 3: Dividing by (3\times7=21) makes the powers 4 and 2.
If the number is (2^5\times3^4\times5^2\times7^7), by which smallest number should it be multiplied to make a perfect cube?
Correct answer: A
Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Powers 5, 4, 2, and 7 must become 6, 6, 3, and 9. Step 3: The smallest multiplier is (2\times3^2\times5\times7^2).
If the number is (2^{10}\times3^8\times5^5\times11^4), by which smallest number should it be divided to make a perfect cube?
Correct answer: A
Step 1: In a perfect cube, exponents are 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\times11).
If (n=2^8\times3^6\times5^2\times7^4), by which number will (n) not be divisible?
Correct answer: A
Step 1: Every prime power of a divisor must be available in the number. Step 2: (n) has power 2 of 5, but (5^3) needs power 3. Step 3: Therefore, (n) is not divisible by (5^3).
If (n=2^9\times3^4\times5^5\times13^2), by which number must (n) be divisible?
Correct answer: A
Step 1: For divisibility, the exponents of the divisor must not exceed those in the number. Step 2: In the first option, all exponents are less than or equal to those in (n). Step 3: Therefore, (n) must be divisible by that number.
If a number has prime factorisation (2^8\times3^6\times5^4\times7^3\times11^2), how many prime factors does it have with repetition?
Correct answer: A
Step 1: To count with repetition, add the exponents. Step 2: (8+6+4+3+2=23). Step 3: Counting only bases and counting with repetition are different.
If a number has prime factorisation (2^6\times3^5\times5^4\times7^3\times11^2\times13), how many distinct prime factors does it have?
Correct answer: A
Step 1: While counting distinct primes, only bases are counted. Step 2: The bases are 2, 3, 5, 7, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 6.
If (a=2^5\times3^3\times5^2\times11) and (b=2^4\times3^5\times7\times11^2), what will be the power of 11 in (ab)?
Correct answer: A
Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 11 in (a) is 1 and in (b) is 2. Step 3: In (ab), the power of 11 will be (1+2=3).
If (x=2^6\times3^4\times7^2) and (y=2^3\times3^5\times5\times7), what will be the power of 3 in (xy)?
Correct answer: A
Step 1: Powers with the same base 3 are added in multiplication. Step 2: The power of 3 in (x) is 4 and in (y) is 5. Step 3: The total power will be (4+5=9).
Which option gives the correct value of (2^6\times3^4\times11^2)?
Correct answer: A
Step 1: Calculate (2^6=64), (3^4=81), and (11^2=121). Step 2: (64\times81\times121=627264). Step 3: Solve all three powers separately first.
Which option gives the correct value of (2^4\times3^3\times5^2\times7^2)?
Correct answer: A
Step 1: Calculate (2^4=16), (3^3=27), (5^2=25), and (7^2=49). Step 2: (16\times27\times25\times49=529200). Step 3: Do the multiplication step by step.
Which option gives only the final prime factorisation?
Correct answer: A
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: 32, 81, 25, 405, 35, and 160 are composite, so they are not final forms.
Which option has incomplete prime factorisation?
Correct answer: A
Step 1: In an incomplete form, composite bases remain. Step 2: 81 and 121 are composite bases. Step 3: (2^6\times81\times121) must be changed into (2^6\times3^4\times11^2).
If (2^4\times3^5\times5^2\times7) is the prime factorisation of a number, what is the number?
Correct answer: A
Step 1: Take (2^4=16), (3^5=243), (5^2=25), and (7). Step 2: (16\times243\times25\times7=680400). Step 3: Solving powers first helps find the correct option quickly.
If (2^3\times3^3\times5^2\times7\times11) is the prime factorisation of a number, what is the number?
Correct answer: A
Step 1: Calculate (2^3=8), (3^3=27), and (5^2=25). Step 2: (8\times27\times25\times7\times11=415800). Step 3: When there are many factors, multiply in small groups.
What is the correct prime factorisation of 582120?
Correct answer: A
Step 1: Write the number using prime-base groups. Step 2: (16\times27\times25\times49\times11=2^4\times3^3\times5^2\times7^2\times11). Step 3: Avoid decimal-based options and write prime bases.
Which is the prime factorisation of 254016?
Correct answer: A
Step 1: Write (254016=64\times3969). Step 2: (64=2^6) and (3969=3^4\times7^2), so (254016=2^6\times3^4\times7^2). Step 3: Convert 3969 into prime powers.
What is the correct prime factorisation of 2138400?
Correct answer: A
Step 1: Write (2138400=32\times66825). Step 2: (66825=3^5\times5^2\times11), so (2138400=2^5\times3^5\times5^2\times11). Step 3: Give 66825 its complete prime form.
Why will (64\times3969) not be considered the final answer in prime factorisation?
Correct answer: A
Step 1: In final prime factorisation, every base should be prime. Step 2: (64=2^6) and (3969=3^4\times7^2). Step 3: Therefore, the final form is (2^6\times3^4\times7^2).
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