Find the number of factors of 9600.

Step-by-Step Solution

To find the total number of factors for any integer, use its prime factorization.

1. Perform Prime Factorization:

   • Break down 9600 into prime factors:

   • 9600 = 96 * 100

   • 96 = 32 * 3 = 2^5 * 3

   • 100 = 10 * 10 = (2 * 5) * (2 * 5) = 2^2 * 5^2

   • Combine them: 9600 = (2^5 * 3) * (2^2 * 5^2)

   • 9600 = 2^7 * 3^1 * 5^2

2. Apply the Factor Counting Formula:

   • If a number is expressed as n = (p^a) * (q^b) * (r^c)..., the total number of factors is (a + 1) * (b + 1) * (c + 1)...

   • Here, the exponents are 7, 1, and 2.

   • Total factors = (7 + 1) * (1 + 1) * (2 + 1)

   • Total factors = 8 * 2 * 3

   • Total factors = 48.