The number of factors of 36 is

Options

A. 6
B. 7
C. 8
D. 9

Answer

To find the number of factors of 36, we can follow these steps:

### Step 1: Find the prime factorization of 36.
To do this, we divide 36 by the smallest prime numbers until we reach 1.

- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

### Step 2: Use the formula to find the number of factors.
The formula to find the number of factors from the prime factorization is:
If \( n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k} \), then the number of factors \( d(n) \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

For 36:
- \( e_1 = 2 \) (for the prime number 2)
- \( e_2 = 2 \) (for the prime number 3)

So, we calculate:
\[ d(36) = (2 + 1)(2 + 1) = 3 \times 3 = 9 \]

### Step 3: Conclusion
The number of factors of 36 is 9.