Alana noticed that a number like 8 = 2 ! (i.e. 𝑝𝑟𝑖𝑚𝑒 ! ) has exactly four factors: 1, 2, 2 ! , and 2 ! . What other numbers are of this form have exactly four factors? Do all the perfect cubes have exactly four factors? Is it true that all numbers that have exactly four factors are perfect cubes? Give reasoning in simple words.

To understand the questions fully, we need to dissect the concept of a number having exactly four factors and whether it’s related to perfect cubes.

### Numbers with Exactly Four Factors

A number will have exactly four factors if and only if it can be expressed in one of the following ways:
1. \( p^3 \), where \( p \) is a prime number.
2. \( p \times q \), where \( p \) and \( q \) are distinct prime numbers.

**Explanation**:
- **For \( p^3 \)**: The factors are \( 1, p, p^2, \) and \( p^3 \).
- **For \( p \times q \)**: The factors are \( 1, p, q, \) and \( p \times q \).

### Perfect Cubes and Factors

Perfect cubes have the form \( n^3 \), where \( n \) is any integer. If \( n \) is a prime number, then as discussed, \( n^3 \) has exactly four factors. However, if \( n \) is not prime, \( n^3 \) will generally have more than four factors because the prime factorization of \( n \) itself will contribute to a greater number of combinations of factors.

**Example**: 
- If \( n = 2 \), then \( n^3 = 8 \) has factors \( 1, 2, 4, 8 \).
- If \( n = 4 \), then \( n^3 = 64 \) has factors \( 1, 2, 4, 8, 16, 32, 64 \).

### Conclusion

Not all perfect cubes have exactly four factors—only those of the form \( p^3 \) (where \( p \) is prime) do. 

Furthermore, not all numbers that have exactly four factors are perfect cubes; they could also be the product of two distinct primes. Thus, the claim that all numbers with exactly four factors are perfect cubes is false.

### Key Takeaways
- Perfect cubes are numbers like \( 8, 27, 64, \ldots \)
- A number has exactly four factors if it’s either the cube of a prime or the product of two distinct primes.
- Perfect cubes can have more than four factors if their base is not a prime.

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Do you want more details, or have any questions? Here are some questions to expand your understanding:

1. What are some examples of numbers with four factors?
2. How can the prime factorization of a number determine its number of factors?
3. What are the properties of prime numbers?
4. Can numbers be expressed as the product of more than two distinct primes and still have four factors?
5. What are the factors of non-prime perfect cubes like \( 27 \) or \( 64 \)?
6. How does the Fundamental Theorem of Arithmetic relate to this topic?
7. What are the differences in factors between powers of primes versus composite numbers?
8. Can you list numbers between 1 and 100 with exactly four factors?

**Tip:** To find the number of factors of any number, determine its prime factorization and use the formula involving the exponents of these primes.

Explore the relationship between numbers with exactly four factors and perfect cubes. Learn how prime factorization and the properties of perfect cubes determine the number of factors a number has. Suitable for students in Grades 7-9.

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