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 3+5+7+2+1=183 + 5 + 7 + 2 + 1 = 18, which is divisible by 3. So, 35721 is divisible by 3.

35721÷3=1190735721 \div 3 = 11907

  • Sum of digits of 11907 is 1+1+9+0+7=181 + 1 + 9 + 0 + 7 = 18, which is also divisible by 3.

11907÷3=396911907 \div 3 = 3969

  • Sum of digits of 3969 is 3+9+6+9=273 + 9 + 6 + 9 = 27, divisible by 3.

3969÷3=13233969 \div 3 = 1323

  • Sum of digits of 1323 is 1+3+2+3=91 + 3 + 2 + 3 = 9, divisible by 3.

1323÷3=4411323 \div 3 = 441

  • Sum of digits of 441 is 4+4+1=94 + 4 + 1 = 9, divisible by 3.

441÷3=147441 \div 3 = 147

  • Sum of digits of 147 is 1+4+7=121 + 4 + 7 = 12, divisible by 3.

147÷3=49147 \div 3 = 49

  • 49 is divisible by 7.

49÷7=749 \div 7 = 7

  • 7 is a prime number.

Step 2: Prime factorization result

We have:

35721=36×7235721 = 3^6 \times 7^2

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 363^6 is already a perfect cube.
  • The factor 727^2 is not a perfect cube. To make it a perfect cube, we need to divide 727^2 by 727^2, 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:

  1. What are the rules for identifying perfect cubes from prime factorizations?
  2. How can we generalize this method for any number to find if it can be made a perfect square or higher power?
  3. What is the difference between making a number a perfect square and a perfect cube?
  4. How can prime factorization help in solving problems related to least common multiples?
  5. 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