How many positive factors of 288 are there?

Options

A. 18
B. 15
C. 10
D. 16

Answer

To find the number of positive factors of 288, we will follow these steps:

### Step 1: Prime Factorization of 288
We start by breaking down 288 into its prime factors.

1. Divide 288 by 2 (the smallest prime number):
- \( 288 \div 2 = 144 \)
2. Divide 144 by 2:
- \( 144 \div 2 = 72 \)
3. Divide 72 by 2:
- \( 72 \div 2 = 36 \)
4. Divide 36 by 2:
- \( 36 \div 2 = 18 \)
5. Divide 18 by 2:
- \( 18 \div 2 = 9 \)
6. Now, 9 is not divisible by 2, so we move to the next prime number, which is 3:
- \( 9 \div 3 = 3 \)
7. Finally, divide 3 by 3:
- \( 3 \div 3 = 1 \)

So, the prime factorization of 288 is:
\[ 288 = 2^5 \times 3^2 \]

### Step 2: Use the Formula to Find the Number of Positive Factors
The formula to find the number of positive factors from the prime factorization \( p^a \times q^b \) is:
\[ (a + 1)(b + 1) \]

In our case:
- For \( 2^5 \), \( a = 5 \)
- For \( 3^2 \), \( b = 2 \)

### Step 3: Calculate the Number of Positive Factors
Now we apply the formula:
\[ (5 + 1)(2 + 1) = 6 \times 3 = 18 \]

### Conclusion
The total number of positive factors of 288 is **18**.