Understanding Factors
To find the number of different factors of a number, we first need to determine its prime factorization.
Prime Factorization of 75,600
- Begin by breaking down 75,600 into its prime factors.
- 75,600 can be expressed as:
- 75,600 = 75.6 x 1,000
- 75 = 3 x 5²
- 1,000 = 10³ = (2 x 5)³ = 2³ x 5³
- Thus, combining these factors gives us:
- 75,600 = 3¹ x 5² x 2³
Now, let's combine all prime factors:
- 75,600 = 2³ x 3¹ x 5⁴
Calculating the Number of Factors
To determine the number of factors from the prime factorization, use the formula:
- If the prime factorization is p¹ x q² x r³, then the number of factors is (e1 + 1)(e2 + 1)(e3 + 1), where e1, e2, and e3 are the exponents in the prime factorization.
In this case:
- Exponents from 75,600 are 3 for 2, 1 for 3, and 4 for 5.
Now, apply the formula:
- (3 + 1)(1 + 1)(4 + 1)
Final Calculation
- This simplifies to:
- 4 x 2 x 5 = 40
Conclusion
Thus, the number of different factors of 75,600 is 40.