What is the smallest number by which 35721 must be divided so that the quotient is a perfect cube

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

\[
35721 \div 3 = 11907
\]

- Sum of digits of 11907 is \( 1 + 1 + 9 + 0 + 7 = 18 \), which is also divisible by 3.

\[
11907 \div 3 = 3969
\]

- Sum of digits of 3969 is \( 3 + 9 + 6 + 9 = 27 \), divisible by 3.

\[
3969 \div 3 = 1323
\]

- Sum of digits of 1323 is \( 1 + 3 + 2 + 3 = 9 \), divisible by 3.

\[
1323 \div 3 = 441
\]

- Sum of digits of 441 is \( 4 + 4 + 1 = 9 \), divisible by 3.

\[
441 \div 3 = 147
\]

- Sum of digits of 147 is \( 1 + 4 + 7 = 12 \), divisible by 3.

\[
147 \div 3 = 49
\]

- 49 is divisible by 7.

\[
49 \div 7 = 7
\]

- 7 is a prime number.

### Step 2: Prime factorization result
We have:

\[
35721 = 3^6 \times 7^2
\]

### 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 \(3^6\) is already a perfect cube.
- The factor \(7^2\) is not a perfect cube. To make it a perfect cube, we need to divide \(7^2\) by \(7^2\), 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**.

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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.

What is the smallest number by which 35721 must be divided so that the quotient is a perfect cube?

Explore how to find the smallest number by which 35721 must be divided to get a perfect cube. This solution involves prime factorization and understanding how exponents need to be multiples of 3 for a number to be a perfect cube. Suitable for Grades 9-12.

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