Find the total number of factors of 576.

Answer

To find the total number of factors of 576, we will follow these steps:

### Step 1: Prime Factorization of 576
We need to express 576 as a product of its prime factors. We can start dividing 576 by the smallest prime number, which is 2.

- 576 ÷ 2 = 288
- 288 ÷ 2 = 144
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 576 is:
\[ 576 = 2^6 \times 3^2 \]

### Step 2: Use the Formula for the Total Number of Factors
The total number of factors of a number can be found using the formula:
If \( n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k} \), then the total number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

In our case:
- For \( 2^6 \), \( e_1 = 6 \)
- For \( 3^2 \), \( e_2 = 2 \)

Using the formula:
\[ T(576) = (6 + 1)(2 + 1) \]
\[ T(576) = 7 \times 3 \]

### Step 3: Calculate the Total Number of Factors
Now we can calculate:
\[ T(576) = 7 \times 3 = 21 \]

Thus, the total number of factors of 576 is **21**.