If `N=2^3xx3^2` then what are the positive factors of N?

Options

A. A)5
B. B)8
C. C)12
D. C)20

Answer

To find the positive factors of \( N = 2^3 \times 3^2 \), we can follow these steps:

### Step 1: Identify the prime factorization
The number \( N \) is given as \( 2^3 \times 3^2 \). Here, the prime factors are 2 and 3.

### Step 2: Determine the powers of the prime factors
In the expression \( 2^3 \times 3^2 \):
- The power of 2 is 3.
- The power of 3 is 2.

### Step 3: Use the formula for finding the number of factors
To find the total number of positive factors of a number given its prime factorization \( p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_n^{e_n} \), we use the formula:
\[ \text{Total Factors} = (e_1 + 1)(e_2 + 1) \ldots (e_n + 1) \]
where \( e_i \) is the exponent of the prime factor \( p_i \).

### Step 4: Apply the formula
For our number \( N = 2^3 \times 3^2 \):
- For the prime factor 2, the exponent \( e_1 = 3 \). Therefore, \( e_1 + 1 = 3 + 1 = 4 \).
- For the prime factor 3, the exponent \( e_2 = 2 \). Therefore, \( e_2 + 1 = 2 + 1 = 3 \).

Now, we can calculate the total number of factors:
\[ \text{Total Factors} = (3 + 1)(2 + 1) = 4 \times 3 = 12 \]

### Conclusion
The total number of positive factors of \( N \) is **12**.

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