The number of factors 348 is

Options

A. 10
B. 9
C. 11
D. 12

Answer

To find the number of factors of the number 348, we can follow these steps:

### Step 1: Prime Factorization of 348
First, we need to find the prime factorization of 348. We can do this by dividing 348 by the smallest prime numbers until we reach 1.

1. Divide 348 by 2 (the smallest prime number):
\[ 348 \div 2 = 174 \]
2. Divide 174 by 2:
\[ 174 \div 2 = 87 \]
3. 87 is not divisible by 2, so we move to the next prime number, which is 3:
\[ 87 \div 3 = 29 \]
4. 29 is a prime number, so we stop here.

Thus, the prime factorization of 348 is:
\[ 348 = 2^2 \times 3^1 \times 29^1 \]

### 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:
\[ \text{Number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \(e_1, e_2, e_3, \ldots\) are the exponents in the prime factorization.

From our factorization \(2^2 \times 3^1 \times 29^1\):
- The exponent of 2 is 2, so \(e_1 = 2\).
- The exponent of 3 is 1, so \(e_2 = 1\).
- The exponent of 29 is 1, so \(e_3 = 1\).

Now, we apply the formula:
\[ \text{Number of factors} = (2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 \]

### Step 3: Calculate the Total
Now we calculate:
\[ 3 \times 2 = 6 \]
\[ 6 \times 2 = 12 \]

### Conclusion
The total number of factors of 348 is 12.

### Final Answer
Thus, the number of factors of 348 is **12**.

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