Total number of factors of 36 is :

Options

A. 4
B. 9
C. 6
D. None of these

Answer

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

### Step 1: Prime Factorization
First, we need to find the prime factorization of 36.

36 can be divided by 2:
- 36 ÷ 2 = 18
- 18 can also be divided by 2:
- 18 ÷ 2 = 9
- Now, 9 can be divided by 3:
- 9 ÷ 3 = 3
- Finally, 3 is a prime number.

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

### Step 2: Use the Formula for Total Number of Factors
To find the total number of factors from the prime factorization, we use the formula:
If a number \( n \) can be expressed as \( p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \), where \( p_1, p_2, \ldots, p_k \) are prime factors and \( e_1, e_2, \ldots, e_k \) are their respective powers, then the total number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]

### Step 3: Apply the Formula
For 36, we have:
- The prime factor \( 2 \) has an exponent of \( 2 \) (from \( 2^2 \)).
- The prime factor \( 3 \) also has an exponent of \( 2 \) (from \( 3^2 \)).

Now, applying the formula:
\[ T(36) = (2 + 1)(2 + 1) = 3 \times 3 = 9 \]

### Conclusion
Thus, the total number of factors of 36 is **9**.

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