How many positive factors of 24 are there?

Options

A. 3
B. 4
C. 6
D. 8

Answer

To find the number of positive factors of 24, we can follow these steps:

### Step 1: Prime Factorization of 24
First, we need to perform the prime factorization of 24.

24 can be divided by 2:
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is a prime number.

So, the prime factorization of 24 is:
\[ 24 = 2^3 \times 3^1 \]

### Step 2: Use the Formula for Counting Factors
To find the total number of positive factors of a number, we use the formula based on its prime factorization. If a number is expressed as:
\[ p_1^{e_1} \times p_2^{e_2} \times ... \times p_n^{e_n} \]
where \( p_1, p_2, ..., p_n \) are the prime factors and \( e_1, e_2, ..., e_n \) are their respective powers, then the total number of positive factors is given by:
\[ (e_1 + 1)(e_2 + 1)...(e_n + 1) \]

### Step 3: Apply the Formula
For 24, we have:
- \( e_1 = 3 \) (for the prime factor 2)
- \( e_2 = 1 \) (for the prime factor 3)

Now, we apply the formula:
\[ (3 + 1)(1 + 1) = 4 \times 2 = 8 \]

### Step 4: Conclusion
Thus, the total number of positive factors of 24 is **8**.