How many positive factors are there of 360?

Options

A. a. 24
B. b. 28
C. c. 18
D. d. 30

Answer

To find the number of positive factors of 360, we will follow these steps:

### Step 1: Prime Factorization of 360
We start by performing the prime factorization of 360.

- Divide 360 by 2:
\( 360 \div 2 = 180 \)
- Divide 180 by 2:
\( 180 \div 2 = 90 \)
- Divide 90 by 2:
\( 90 \div 2 = 45 \)
- Divide 45 by 3:
\( 45 \div 3 = 15 \)
- Divide 15 by 3:
\( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.

So, the prime factorization of 360 is:
\[ 360 = 2^3 \times 3^2 \times 5^1 \]

### Step 2: Use the Formula to Find the Number of Factors
To find the total number of positive factors, we use the formula:
If \( n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \), then the number of factors \( d(n) \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

In our case, we have:
- For \( 2^3 \), \( e_1 = 3 \)
- For \( 3^2 \), \( e_2 = 2 \)
- For \( 5^1 \), \( e_3 = 1 \)

Now, we calculate:
\[ d(360) = (3 + 1)(2 + 1)(1 + 1) \]
\[ d(360) = 4 \times 3 \times 2 \]

### Step 3: Calculate the Total Number of Factors
Now we perform the multiplication:
\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]

Thus, the total number of positive factors of 360 is **24**.

### Final Answer
The number of positive factors of 360 is **24**.

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