Find the total number of factors of 1680.

Options

A. 40
B. 50
C. 60
D. 30

Answer

To find the total number of factors of the number 1680, we will follow these steps:

### Step 1: Prime Factorization of 1680
We start by finding the prime factorization of 1680.

1. Divide by 2:
- 1680 ÷ 2 = 840
- 840 ÷ 2 = 420
- 420 ÷ 2 = 210
- 210 ÷ 2 = 105 (we can no longer divide by 2)

2. Now, we divide by 3:
- 105 ÷ 3 = 35 (we can no longer divide by 3)

3. Next, we divide by 5:
- 35 ÷ 5 = 7 (we can no longer divide by 5)

4. Finally, we are left with 7, which is a prime number.

Thus, the prime factorization of 1680 is:
\[ 1680 = 2^4 \times 3^1 \times 5^1 \times 7^1 \]

### Step 2: Use the Formula for Total Number of Factors
To find the total number of factors, we use the formula:
If a number \( n \) can be expressed as:
\[ n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \times \ldots \]
Then the total number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]

For our factorization \( 1680 = 2^4 \times 3^1 \times 5^1 \times 7^1 \):
- \( e_1 = 4 \) (for \( 2 \))
- \( e_2 = 1 \) (for \( 3 \))
- \( e_3 = 1 \) (for \( 5 \))
- \( e_4 = 1 \) (for \( 7 \))

### Step 3: Calculate the Total Number of Factors
Now we apply the formula:
\[ T(1680) = (4 + 1)(1 + 1)(1 + 1)(1 + 1) \]
\[ T(1680) = 5 \times 2 \times 2 \times 2 \]

Calculating this step-by-step:
1. \( 5 \times 2 = 10 \)
2. \( 10 \times 2 = 20 \)
3. \( 20 \times 2 = 40 \)

Thus, the total number of factors of 1680 is **40**.

### Final Answer
The total number of factors of 1680 is **40**.

---