Find the total number of factors of 540.

Options

A. 24
B. 20
C. 30
D. None of these

Answer

To find the total number of factors of 540, we will follow these steps:

### Step 1: Prime Factorization of 540
We start by finding the prime factors of 540.

1. **Divide by 2**:
\( 540 \div 2 = 270 \)
So, we have \( 2^1 \).

2. **Divide by 2 again**:
\( 270 \div 2 = 135 \)
So, we have \( 2^2 \).

3. **Divide by 3**:
\( 135 \div 3 = 45 \)
So, we have \( 3^1 \).

4. **Divide by 3 again**:
\( 45 \div 3 = 15 \)
So, we have \( 3^2 \).

5. **Divide by 3 again**:
\( 15 \div 3 = 5 \)
So, we have \( 3^3 \).

6. **Divide by 5**:
\( 5 \div 5 = 1 \)
So, we have \( 5^1 \).

Combining these, we get the prime factorization of 540:
\[ 540 = 2^2 \times 3^3 \times 5^1 \]

### Step 2: Use the Formula to Find Total Number of Factors
The formula to find the total number of factors from the prime factorization is:
If \( n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \), then the total number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

For our case:
- \( e_1 = 2 \) (for \( 2^2 \))
- \( e_2 = 3 \) (for \( 3^3 \))
- \( e_3 = 1 \) (for \( 5^1 \))

### Step 3: Calculate the Total Number of Factors
Now we apply the formula:
\[ T(540) = (2 + 1)(3 + 1)(1 + 1) \]

Calculating each term:
- \( 2 + 1 = 3 \)
- \( 3 + 1 = 4 \)
- \( 1 + 1 = 2 \)

Now multiply these results:
\[ T(540) = 3 \times 4 \times 2 \]

Calculating:
- \( 3 \times 4 = 12 \)
- \( 12 \times 2 = 24 \)

### Final Answer
Thus, the total number of factors of 540 is **24**.

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