Find the total number of factors of 1420 ?

Options

A. 12
B. 13
C. 14
D. 15

Answer

To find the total number of factors of 1420, we will follow these steps:

### Step 1: Prime Factorization of 1420
We start by finding the prime factorization of 1420. We can do this by dividing 1420 by the smallest prime numbers.

1. **Divide by 2**:
\[ 1420 \div 2 = 710 \]
2. **Divide by 2 again**:
\[ 710 \div 2 = 355 \]
3. **Divide by 5**:
\[ 355 \div 5 = 71 \]
4. **71 is a prime number**.

So, the prime factorization of 1420 is:
\[ 1420 = 2^2 \times 5^1 \times 71^1 \]

### Step 2: Use the Formula for Total Number of Factors
To find the total number of factors, we use the formula:
\[ \text{Total number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \(e_1, e_2, e_3, \ldots\) are the powers of the prime factors.

From our prime factorization:
- The power of 2 is 2 (i.e., \(e_1 = 2\))
- The power of 5 is 1 (i.e., \(e_2 = 1\))
- The power of 71 is 1 (i.e., \(e_3 = 1\))

Now we can apply the formula:
\[ \text{Total number of factors} = (2 + 1)(1 + 1)(1 + 1) \]

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

Now multiply these together:
\[ \text{Total number of factors} = 3 \times 2 \times 2 = 12 \]

### Final Answer
The total number of factors of 1420 is **12**.

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