How many positive factors of 72 are there?

Options

A. 6
B. 10
C. 8
D. 12

Answer

To find the number of positive factors of 72, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of 72. We can do this by dividing 72 by the smallest prime numbers until we reach 1.

- 72 is even, so we start with 2:
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- Now, 9 is not divisible by 2, so we move to the next prime number, which is 3:
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

Thus, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

### Step 2: Use the Formula for Counting Factors
To find the total number of positive factors from the prime factorization, we use the formula:
If \( n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \), then the number of positive factors \( d(n) \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]

For 72:
- The exponents from the prime factorization \( 2^3 \) and \( 3^2 \) are 3 and 2, respectively.

So, we calculate:
\[ d(72) = (3 + 1)(2 + 1) \]
\[ d(72) = 4 \times 3 = 12 \]

### Conclusion
Therefore, the number of positive factors of 72 is **12**.

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