Find the number of factors of factors of the following :
512

Answer

To find the number of factors of the number 512, we will follow these steps:

### Step 1: Prime Factorization of 512
We start by finding the prime factorization of 512. We can do this by dividing 512 by the smallest prime number, which is 2.

- 512 ÷ 2 = 256
- 256 ÷ 2 = 128
- 128 ÷ 2 = 64
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

So, we can express 512 as:
\[ 512 = 2^9 \]

### Step 2: Use the Formula to Find the Number of Factors
The formula to find the number of factors of a number based on its prime factorization is:
If a number \( n \) can be expressed as \( p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k} \), then the total number of factors \( N \) is given by:
\[ N = (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

In our case, since \( 512 = 2^9 \), we have:
- \( e_1 = 9 \)

Thus, the total number of factors \( N \) is:
\[ N = (9 + 1) = 10 \]

### Conclusion
Therefore, the total number of factors of 512 is **10**.

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