There are different methods to find the number of factors of a given number. One common way is to factorize the number into its prime factors and then use the formula:

The number of factors of a number = (a+1)(b+1)(c+1)...(n+1) where a, b, c, ..., n are the powers of the prime factors in the factorization.



To find the prime factorization of 75600, we can use the following steps:

1. Divide the number by the smallest prime factor, which is 2. We get 2 x 37800.
2. Divide 37800 by 2. We get 2 x 18900.
3. Divide 18900 by 2. We get 2 x 9450.
4. Divide 9450 by 3. We get 2 x 3 x 3150.
5. Divide 3150 by 2. We get 2 x 3 x 5 x 1050.
6. Divide 1050 by 2. We get 2 x 3 x 5 x 7 x 15.
7. We cannot divide 15 further since it is not divisible by any prime number.

Therefore, the prime factorization of 75600 is 2^4 x 3^3 x 5^2 x 7.



Using the formula mentioned above, we can find the number of factors of 75600 as:

Number of factors = (4+1)(3+1)(2+1)(1+1) = 5 x 4 x 3 x 2 = 120.

Therefore, the number of different factors of 75600 is 120.