How many positive factors of 444 are there?

Options

A. 6
B. 8
C. 9
D. 12

Answer

To determine how many positive factors of 444 there are, we can follow these steps:

### Step 1: Prime Factorization of 444
First, we need to find the prime factorization of 444. We can start by dividing by the smallest prime number, which is 2.

- **Divide 444 by 2:**
\[ 444 \div 2 = 222 \]

- **Divide 222 by 2:**
\[ 222 \div 2 = 111 \]

- **Now, divide 111 by 3 (the next smallest prime):**
\[ 111 \div 3 = 37 \]

- **Finally, 37 is a prime number.**

So, the prime factorization of 444 is:
\[ 444 = 2^2 \times 3^1 \times 37^1 \]

### Step 2: Use the Formula to Find the Number of Factors
To find the total number of positive factors, we use the formula based on the prime factorization. If a number \( n \) can be expressed as:
\[ n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \ldots \]
where \( p_1, p_2, p_3 \) are prime factors and \( e_1, e_2, e_3 \) are their respective powers, then the total number of positive factors \( T \) is given by:
\[ T = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]

For our case:
- The powers of the prime factors from the factorization \( 2^2 \times 3^1 \times 37^1 \) are:
- \( e_1 = 2 \) (for 2)
- \( e_2 = 1 \) (for 3)
- \( e_3 = 1 \) (for 37)

### Step 3: Calculate the Total Number of Factors
Now we apply the formula:
\[ T = (2 + 1)(1 + 1)(1 + 1) \]
Calculating each term:
- \( 2 + 1 = 3 \)
- \( 1 + 1 = 2 \)
- \( 1 + 1 = 2 \)

Now multiply these results:
\[ T = 3 \times 2 \times 2 = 12 \]

### Conclusion
Thus, the total number of positive factors of 444 is **12**.

---