To find all the factors of 384, we will follow these steps:
### Step 1: Prime Factorization of 384
We begin by finding the prime factorization of 384. We can do this by dividing the number by the smallest prime number (2) and continuing the process with the quotient until we reach 1.
1. **Divide by 2**:
- 384 ÷ 2 = 192
2. **Divide by 2**:
- 192 ÷ 2 = 96
3. **Divide by 2**:
- 96 ÷ 2 = 48
4. **Divide by 2**:
- 48 ÷ 2 = 24
5. **Divide by 2**:
- 24 ÷ 2 = 12
6. **Divide by 2**:
- 12 ÷ 2 = 6
7. **Divide by 2**:
- 6 ÷ 2 = 3
8. **Divide by 3**:
- 3 ÷ 3 = 1
Thus, the prime factorization of 384 is:
\[
384 = 2^7 \times 3^1
\]
### Step 2: Finding All Factors
To find all the factors of a number from its prime factorization, we use the formula:
If \( n = p_1^{e_1} \times p_2^{e_2} \), then the total number of factors is given by:
\[
(e_1 + 1)(e_2 + 1)
\]
For 384:
- \( e_1 = 7 \) (for 2)
- \( e_2 = 1 \) (for 3)
Thus, the total number of factors is:
\[
(7 + 1)(1 + 1) = 8 \times 2 = 16
\]
### Step 3: Listing All Factors
To list all factors, we can take all combinations of the prime factors raised to their respective powers. The factors can be calculated as follows:
- From \( 2^0 \) to \( 2^7 \) (which gives us 1, 2, 4, 8, 16, 32, 64, 128)
- From \( 3^0 \) to \( 3^1 \) (which gives us 1, 3)
Now we can multiply these combinations:
1. \( 2^0 \times 3^0 = 1 \)
2. \( 2^1 \times 3^0 = 2 \)
3. \( 2^2 \times 3^0 = 4 \)
4. \( 2^3 \times 3^0 = 8 \)
5. \( 2^4 \times 3^0 = 16 \)
6. \( 2^5 \times 3^0 = 32 \)
7. \( 2^6 \times 3^0 = 64 \)
8. \( 2^7 \times 3^0 = 128 \)
9. \( 2^0 \times 3^1 = 3 \)
10. \( 2^1 \times 3^1 = 6 \)
11. \( 2^2 \times 3^1 = 12 \)
12. \( 2^3 \times 3^1 = 24 \)
13. \( 2^4 \times 3^1 = 48 \)
14. \( 2^5 \times 3^1 = 96 \)
15. \( 2^6 \times 3^1 = 192 \)
16. \( 2^7 \times 3^1 = 384 \)
### Final List of Factors
The factors of 384 are:
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384
