Find the total number of factors of 24.

Answer

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

### Step 1: Prime Factorization of 24
First, we need to express 24 as a product of its prime factors.
- 24 can be divided by 2:
- 24 ÷ 2 = 12
- 12 can be divided by 2:
- 12 ÷ 2 = 6
- 6 can be divided by 2:
- 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 Total Number of Factors
To find the total number of factors, we use 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:
\[ T(n) = (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

For 24, we have:
- \( p_1 = 2 \) with \( e_1 = 3 \)
- \( p_2 = 3 \) with \( e_2 = 1 \)

Applying the formula:
\[ T(24) = (3 + 1)(1 + 1) \]
\[ T(24) = 4 \times 2 \]

### Step 3: Calculate the Total Number of Factors
Now, we calculate:
\[ T(24) = 4 \times 2 = 8 \]

### Conclusion
Thus, the total number of factors of 24 is **8**.

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