How many positive factors of 160 are there?

Options

A. 6
B. 8
C. 5
D. 12

Answer

To find the number of positive factors of 160, we will follow these steps:

### Step 1: Prime Factorization of 160
We start by finding the prime factorization of 160.

- Divide 160 by 2 (the smallest prime number):
\[ 160 \div 2 = 80 \]
- Divide 80 by 2:
\[ 80 \div 2 = 40 \]
- Divide 40 by 2:
\[ 40 \div 2 = 20 \]
- Divide 20 by 2:
\[ 20 \div 2 = 10 \]
- Divide 10 by 2:
\[ 10 \div 2 = 5 \]
- Finally, 5 is a prime number.

So, the prime factorization of 160 is:
\[ 160 = 2^5 \times 5^1 \]

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

In our case, the prime factorization is \(2^5 \times 5^1\):
- The exponent of 2 is 5, so \(e_1 = 5\).
- The exponent of 5 is 1, so \(e_2 = 1\).

Now we apply the formula:
\[ \text{Number of factors} = (5 + 1)(1 + 1) = 6 \times 2 \]

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

Thus, the number of positive factors of 160 is **12**.

### Final Answer
The number of positive factors of 160 is **12**.

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