How many positive factors of 112 are there?

Options

A. 9
B. 10
C. 8
D. 7

Answer

To find the number of positive factors of the number 112, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of 112.

- Start dividing by the smallest prime number, which is 2:
- \( 112 \div 2 = 56 \)
- \( 56 \div 2 = 28 \)
- \( 28 \div 2 = 14 \)
- \( 14 \div 2 = 7 \)
- \( 7 \div 7 = 1 \)

So, we can express 112 as:
\[ 112 = 2^4 \times 7^1 \]

### Step 2: Use the Formula for Counting Factors
To find the total number of positive factors of a number from its prime factorization, we use the formula:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \( e_1, e_2, e_3, \ldots \) are the powers of the prime factors.

In our case, we have:
- For \( 2^4 \), the exponent \( e_1 = 4 \)
- For \( 7^1 \), the exponent \( e_2 = 1 \)

Now, we apply the formula:
\[ (4 + 1)(1 + 1) \]

### Step 3: Calculate the Number of Factors
Now we calculate:
\[ (4 + 1) = 5 \]
\[ (1 + 1) = 2 \]

Now multiply these results:
\[ 5 \times 2 = 10 \]

### Conclusion
Thus, the total number of positive factors of 112 is **10**.