How many positive factors of 120 are there?

Options

A. 12
B. 16
C. 24
D. 36

Answer

To find the number of positive factors of 120, we can follow these steps:

### Step 1: Prime Factorization of 120
First, we need to find the prime factorization of 120. We can do this by dividing 120 by the smallest prime numbers until we reach 1.

- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1

So, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

### Step 2: Use the Formula for Counting Factors
To find the number of positive factors of a number, we can use the formula based on its prime factorization. If a number \( n \) can be expressed as:
\[ n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \]
where \( p_1, p_2, p_3 \) are prime factors and \( e_1, e_2, e_3 \) are their respective powers, then the number of positive factors \( d(n) \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

### Step 3: Apply the Formula
For 120, we have:
- \( e_1 = 3 \) (for the prime factor 2)
- \( e_2 = 1 \) (for the prime factor 3)
- \( e_3 = 1 \) (for the prime factor 5)

Now, we can substitute these values into the formula:
\[ d(120) = (3 + 1)(1 + 1)(1 + 1) \]
\[ d(120) = 4 \times 2 \times 2 \]

### Step 4: Calculate the Total Number of Factors
Now we calculate:
\[ d(120) = 4 \times 2 \times 2 = 16 \]

### Conclusion
Thus, the number of positive factors of 120 is **16**.

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