Find the total number of factors of 240 -

Options

A. 20
B. 25
C. 15
D. 22

Answer

To find the total number of factors of 240, we will follow these steps:

### Step 1: Prime Factorization of 240
First, we need to perform the prime factorization of 240.

- Start by dividing 240 by the smallest prime number, which is 2:
- 240 ÷ 2 = 120
- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- Now, 15 is not divisible by 2, so we move to the next prime number, which is 3:
- 15 ÷ 3 = 5
- Finally, 5 is a prime number itself.

So, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

### Step 2: Use the Formula to Find the Number of Factors
To find the total number of factors from the prime factorization, we use the formula:
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:
\[ T(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

In our case:
- For \( 2^4 \), \( e_1 = 4 \)
- For \( 3^1 \), \( e_2 = 1 \)
- For \( 5^1 \), \( e_3 = 1 \)

Now we calculate:
\[ T(240) = (4 + 1)(1 + 1)(1 + 1) \]

### Step 3: Calculate Each Term
Now we calculate each term:
- \( 4 + 1 = 5 \)
- \( 1 + 1 = 2 \)
- \( 1 + 1 = 2 \)

### Step 4: Multiply the Results
Now we multiply these results together:
\[ T(240) = 5 \times 2 \times 2 \]

### Step 5: Final Calculation
Calculating this gives:
\[ T(240) = 5 \times 2 = 10 \]
\[ 10 \times 2 = 20 \]

Thus, the total number of factors of 240 is **20**.