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? Explain.

### Problem Breakdown:

1. **Understanding the Prime Factorial Form (2!)**:
    - The notation \( p! \) typically represents the factorial of a prime number \( p \).
    - The factorial of \( p = 2 \) is \( 2! = 2 \times 1 = 2 \).
    - Thus, the expression \( 8 = 2! \times 2! \) means that the number 8 is derived from multiplying \( 2! \) by itself, leading to 8 having exactly four factors: 1, 2, 4, and 8.

2. **Exploring Other Numbers with Exactly Four Factors**:
    - For a number \( n \) to have exactly four factors, it must be of a specific form:
      - \( n = p^3 \) (where \( p \) is a prime) or
      - \( n = pq \) (where \( p \) and \( q \) are distinct primes).
  
    - **Case 1: \( n = p^3 \)**
      - \( p^3 \) has exactly four factors: \( 1, p, p^2, p^3 \).
      - Examples: \( 8 = 2^3 \), \( 27 = 3^3 \), \( 125 = 5^3 \).

    - **Case 2: \( n = pq \)**
      - If \( n = pq \), then the factors are \( 1, p, q, pq \), which are exactly four factors.
      - Example: \( 6 = 2 \times 3 \).

3. **Perfect

Explore numbers like 8 = 2! that have exactly four factors. Discover if all perfect cubes have exactly four factors and whether all numbers with four factors are perfect cubes. Learn about prime factorials and their role in determining the number of factors a number has. Suitable for Grades 7-9 students studying number theory and factorization.

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