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The smallest natural number by which 243 must be multiplied to make the product a perfect cube is __________.
- a)3
- b)9
- c)8
- d)7
Correct answer is option 'A'. Can you explain this answer?
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The smallest natural numberby which 243 must be multiplied to make the...
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The smallest natural numberby which 243 must be multiplied to make the...
If we multiply 243by 3 ,we will get,
243×3=729
which is the cube of'7'.
243×3=729
which is the cube of'7'.
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The smallest natural numberby which 243 must be multiplied to make the...
Introduction:
In this question, we need to find the smallest natural number by which 243 must be multiplied to make the product a perfect cube.
Method:
To find the smallest natural number, we need to factorize 243 into its prime factors. Then, we will determine the power of each prime factor and multiply them together. Finally, we will take the cube root of this product to get the smallest natural number.
Factorization of 243:
The prime factorization of 243 can be obtained by dividing it successively by prime numbers until we get only prime factors. Let's perform the factorization:
243 ÷ 3 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Hence, the prime factorization of 243 is 3^5.
Determining the power of each prime factor:
Since we want to make the product a perfect cube, we need to find the power of each prime factor that is not divisible by 3. In this case, the power of 3 is already a multiple of 3 (power of 5). Therefore, we don't need to include 3 in our calculations.
Calculating the smallest natural number:
To calculate the smallest natural number, we need to multiply the prime factors (excluding 3) together and take the cube root of the result.
Prime factors (excluding 3): 1
Product: 1^1 = 1
Cube root of 1: 1
Therefore, the smallest natural number by which 243 must be multiplied to make the product a perfect cube is 1.
Conclusion:
The smallest natural number by which 243 must be multiplied to make the product a perfect cube is 1.
In this question, we need to find the smallest natural number by which 243 must be multiplied to make the product a perfect cube.
Method:
To find the smallest natural number, we need to factorize 243 into its prime factors. Then, we will determine the power of each prime factor and multiply them together. Finally, we will take the cube root of this product to get the smallest natural number.
Factorization of 243:
The prime factorization of 243 can be obtained by dividing it successively by prime numbers until we get only prime factors. Let's perform the factorization:
243 ÷ 3 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Hence, the prime factorization of 243 is 3^5.
Determining the power of each prime factor:
Since we want to make the product a perfect cube, we need to find the power of each prime factor that is not divisible by 3. In this case, the power of 3 is already a multiple of 3 (power of 5). Therefore, we don't need to include 3 in our calculations.
Calculating the smallest natural number:
To calculate the smallest natural number, we need to multiply the prime factors (excluding 3) together and take the cube root of the result.
Prime factors (excluding 3): 1
Product: 1^1 = 1
Cube root of 1: 1
Therefore, the smallest natural number by which 243 must be multiplied to make the product a perfect cube is 1.
Conclusion:
The smallest natural number by which 243 must be multiplied to make the product a perfect cube is 1.
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The smallest natural numberby which 243 must be multiplied to make the product a perfect cube is __________.a)3 b)9c)8 d)7Correct answer is option 'A'. Can you explain this answer?
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The smallest natural numberby which 243 must be multiplied to make the product a perfect cube is __________.a)3 b)9c)8 d)7Correct answer is option 'A'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The smallest natural numberby which 243 must be multiplied to make the product a perfect cube is __________.a)3 b)9c)8 d)7Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The smallest natural numberby which 243 must be multiplied to make the product a perfect cube is __________.a)3 b)9c)8 d)7Correct answer is option 'A'. Can you explain this answer?.
The smallest natural numberby which 243 must be multiplied to make the product a perfect cube is __________.a)3 b)9c)8 d)7Correct answer is option 'A'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The smallest natural numberby which 243 must be multiplied to make the product a perfect cube is __________.a)3 b)9c)8 d)7Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The smallest natural numberby which 243 must be multiplied to make the product a perfect cube is __________.a)3 b)9c)8 d)7Correct answer is option 'A'. Can you explain this answer?.
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