Math Problem Statement
Find the smallest number by which 27783 should be multiplied to get a perfect cube number.
Solution
To find the smallest number by which 27783 should be multiplied to get a perfect cube, we need to factorize 27783 and then adjust its prime factorization so that each exponent is a multiple of 3.
Let's start by factorizing 27783:
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Check for divisibility by 3:
- Sum of the digits of 27783:
- Since 27 is divisible by 3, 27783 is also divisible by 3.
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Factorize 9261:
- Sum of the digits of 9261:
- Since 18 is divisible by 3, 9261 is also divisible by 3.
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Factorize 3087:
- Sum of the digits of 3087:
- Since 18 is divisible by 3, 3087 is also divisible by 3.
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Factorize 1029:
- Sum of the digits of 1029:
- Since 12 is divisible by 3, 1029 is also divisible by 3.
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Factorize 343:
- Sum of the digits of 343:
- 343 is not divisible by 3, so check for divisibility by 7:
So, the prime factorization of 27783 is:
To make this a perfect cube, each exponent in the prime factorization must be a multiple of 3.
- For : The exponent 4 needs to be adjusted to the nearest multiple of 3, which is 6. So we need 2 more factors of 3.
- For : The exponent 3 is already a multiple of 3, so no adjustment is needed.
Therefore, the smallest number by which 27783 should be multiplied to get a perfect cube is:
Thus, the smallest number is .
Would you like more details or have any questions?
Here are 8 related questions:
- What is the prime factorization method?
- How do you check for divisibility by different numbers?
- What is a perfect cube?
- How do you determine the smallest multiplier to achieve a perfect square?
- How can you find the smallest number by which a given number should be divided to get a perfect cube?
- What are the properties of cube numbers?
- Can you explain the exponent rules for cube numbers?
- What is the difference between perfect squares and perfect cubes?
Tip: When factorizing a large number, always start with the smallest prime numbers and check divisibility systematically.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Prime Factorization
Perfect Cubes
Formulas
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Theorems
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Suitable Grade Level
High School
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