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What is the smallest number by which 72 must be multiplied to make it a perfect cube?
- a)2
- b)9
- c)6
- d)3
Correct answer is option 'D'. Can you explain this answer?
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What is the smallest number by which 72 must be multiplied to make it ...
Prime factorising 72, we get, 72 = 2×2×2×3×3 We know, a perfect cube has multiples of 3 as powers of prime factors. Here, number of 2's is 3 and number of 3's is 2. So we need to multiply another 3 in the factorization to make 72 a perfect cube.
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What is the smallest number by which 72 must be multiplied to make it ...
Understanding Perfect Cubes
To determine the smallest number by which 72 must be multiplied to become a perfect cube, we first need to factorize 72 into its prime factors.
Prime Factorization of 72
- 72 can be expressed as:
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- Thus, the complete factorization is:
- 72 = 2^3 × 3^2
Characteristics of Perfect Cubes
- A perfect cube is a number whose prime factors all have exponents that are multiples of 3.
- For example, in a perfect cube, the exponent of each prime factor should ideally be 0, 3, 6, 9, etc.
Analyzing the Factorization
- From the factorization of 72:
- The exponent of 2 is 3 (which is already a multiple of 3).
- The exponent of 3 is 2 (which is not a multiple of 3).
Finding the Required Multiplier
- To make 3^2 a multiple of 3, we need to increase the exponent from 2 to 3.
- Thus, we need one additional factor of 3.
Conclusion
- Therefore, the smallest number we must multiply 72 by is:
- 3 (which corresponds to option 'D').
In summary, multiplying 72 by 3 results in 216, which equals 6^3, confirming it is a perfect cube.
To determine the smallest number by which 72 must be multiplied to become a perfect cube, we first need to factorize 72 into its prime factors.
Prime Factorization of 72
- 72 can be expressed as:
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- Thus, the complete factorization is:
- 72 = 2^3 × 3^2
Characteristics of Perfect Cubes
- A perfect cube is a number whose prime factors all have exponents that are multiples of 3.
- For example, in a perfect cube, the exponent of each prime factor should ideally be 0, 3, 6, 9, etc.
Analyzing the Factorization
- From the factorization of 72:
- The exponent of 2 is 3 (which is already a multiple of 3).
- The exponent of 3 is 2 (which is not a multiple of 3).
Finding the Required Multiplier
- To make 3^2 a multiple of 3, we need to increase the exponent from 2 to 3.
- Thus, we need one additional factor of 3.
Conclusion
- Therefore, the smallest number we must multiply 72 by is:
- 3 (which corresponds to option 'D').
In summary, multiplying 72 by 3 results in 216, which equals 6^3, confirming it is a perfect cube.
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