The number of factors of 540 is

Options

A. 20
B. 22
C. 23
D. 24

Answer

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

### Step 1: Prime Factorization of 540
First, we need to find the prime factorization of 540.

- Divide 540 by the smallest prime number, which is 2:
\[ 540 \div 2 = 270 \]
- Divide 270 by 2 again:
\[ 270 \div 2 = 135 \]
- Now, 135 is not divisible by 2, so we move to the next prime number, which is 3:
\[ 135 \div 3 = 45 \]
- Divide 45 by 3:
\[ 45 \div 3 = 15 \]
- Divide 15 by 3:
\[ 15 \div 3 = 5 \]
- Finally, 5 is a prime number.

Thus, the prime factorization of 540 is:
\[ 540 = 2^2 \times 3^3 \times 5^1 \]

### Step 2: Use the Formula for the Number of Factors
The formula to find the number of factors from the prime factorization is:
\[ \text{Number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \( e_1, e_2, e_3, \ldots \) are the exponents in the prime factorization.

From our factorization \( 2^2 \times 3^3 \times 5^1 \):
- The exponent of 2 is 2, so \( e_1 = 2 \).
- The exponent of 3 is 3, so \( e_2 = 3 \).
- The exponent of 5 is 1, so \( e_3 = 1 \).

Now, substituting these values into the formula:
\[ \text{Number of factors} = (2 + 1)(3 + 1)(1 + 1) \]
\[ = 3 \times 4 \times 2 \]

### Step 3: Calculate the Total
Now we calculate:
\[ 3 \times 4 = 12 \]
\[ 12 \times 2 = 24 \]

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

### Final Answer
The number of factors of 540 is **24**.

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