Math Problem Statement
What is the smallest number by which 35721 must be divided so that the quotient is a perfect cube
Solution
To find the smallest number by which 35721 must be divided so that the quotient is a perfect cube, follow these steps:
Step 1: Prime factorization of 35721
We begin by finding the prime factorization of 35721.
- 35721 is odd, so it is not divisible by 2.
- Sum of digits of 35721 is , which is divisible by 3. So, 35721 is divisible by 3.
- Sum of digits of 11907 is , which is also divisible by 3.
- Sum of digits of 3969 is , divisible by 3.
- Sum of digits of 1323 is , divisible by 3.
- Sum of digits of 441 is , divisible by 3.
- Sum of digits of 147 is , divisible by 3.
- 49 is divisible by 7.
- 7 is a prime number.
Step 2: Prime factorization result
We have:
Step 3: Making the quotient a perfect cube
For a number to be a perfect cube, the exponent of each prime factor in its prime factorization must be a multiple of 3.
- The factor is already a perfect cube.
- The factor is not a perfect cube. To make it a perfect cube, we need to divide by , i.e., we divide 35721 by 49.
Step 4: Conclusion
The smallest number by which 35721 must be divided so that the quotient is a perfect cube is 49.
Would you like further details or have any questions?
Here are 5 related questions to expand on this topic:
- What are the rules for identifying perfect cubes from prime factorizations?
- How can we generalize this method for any number to find if it can be made a perfect square or higher power?
- What is the difference between making a number a perfect square and a perfect cube?
- How can prime factorization help in solving problems related to least common multiples?
- How would the process change if the number had more distinct prime factors?
Tip: Always break down the prime factorization step by step to ensure no errors in complex factorization problems.
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Math Problem Analysis
Mathematical Concepts
Prime Factorization
Perfect Cubes
Exponents
Formulas
Prime factorization of numbers
Condition for perfect cubes: exponents must be multiples of 3
Theorems
Fundamental Theorem of Arithmetic
Suitable Grade Level
Grades 9-12
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