Prime Factorization of 4500
To determine the odd, even perfect square, and perfect cube factors of 4500, we first need its prime factorization.
4500 can be factored as follows:
- 4500 = 45 × 100
- 45 = 5 × 9 = 5 × 3²
- 100 = 10² = (2 × 5)² = 2² × 5²
Thus, the complete factorization is:
4500 = 2² × 3² × 5³

Finding Perfect Square Factors
Perfect square factors must have even powers for all prime factors.
- In 2², the even power is 0 or 2.
- In 3², the even power is 0 or 2.
- In 5³, the even power is 0, 2.
Calculating combinations:
- Choices for 2: 2 options (0, 2)
- Choices for 3: 2 options (0, 2)
- Choices for 5: 2 options (0, 2)
Total perfect square factors = 2 × 2 × 2 = 8.

Finding Perfect Cube Factors
Perfect cube factors must have powers that are multiples of 3.
- In 2², the only choice is 0.
- In 3², the only choice is 0.
- In 5³, choices are 0 or 3.
Calculating combinations:
- Choices for 2: 1 option (0)
- Choices for 3: 1 option (0)
- Choices for 5: 2 options (0, 3)
Total perfect cube factors = 1 × 1 × 2 = 2.

Identifying Odd and Even Factors
- Odd perfect square factors: Consider only 3² and 5³. Total = 2 (0, 2 for 3) × 2 (0, 2 for 5) = 4.
- Even perfect square factors: Must include 2². Total = 2 (0, 2 for 2) × 2 (0, 2 for 3) × 2 (0, 2 for 5) = 4.
In summary:
- Odd perfect square factors: 4
- Even perfect square factors: 4
- Perfect cube factors: 2