If `N=12^(3)xx3^(4)xx5^(2)`, then the total number of even factors of N is

Options

A. 25
B. 121
C. 144
D. 84

Answer

To find the total number of even factors of \( N = 12^3 \times 3^4 \times 5^2 \), we can follow these steps:

### Step 1: Prime Factorization of \( N \)

First, we need to express \( 12^3 \) in terms of its prime factors.

Since \( 12 = 3 \times 2^2 \), we can write:

\[ 12^3 = (3 \times 2^2)^3 = 3^3 \times 2^6 \]

Now substituting this back into \( N \):

\[ N = 12^3 \times 3^4 \times 5^2 = (3^3 \times 2^6) \times 3^4 \times 5^2 \]

### Step 2: Combine Like Terms

Next, we combine the powers of the same base:

\[ N = 3^{3+4} \times 2^6 \times 5^2 = 3^7 \times 2^6 \times 5^2 \]

### Step 3: Total Number of Factors

The formula for finding the total number of factors of a number given its prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \) is:

\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

For \( N = 3^7 \times 2^6 \times 5^2 \), the exponents are \( 7, 6, \) and \( 2 \). Thus, the total number of factors is:

\[ (7 + 1)(6 + 1)(2 + 1) = 8 \times 7 \times 3 \]

Calculating this gives:

\[ 8 \times 7 = 56 \]
\[ 56 \times 3 = 168 \]

### Step 4: Total Number of Odd Factors

To find the number of even factors, we first need to calculate the number of odd factors. Odd factors can only be formed using the odd prime factors, which are \( 3^7 \) and \( 5^2 \).

Using the same formula for the odd factors:

\[ (7 + 1)(2 + 1) = 8 \times 3 = 24 \]

### Step 5: Total Number of Even Factors

Finally, we can find the total number of even factors by subtracting the number of odd factors from the total number of factors:

\[ \text{Total even factors} = \text{Total factors} - \text{Odd factors} = 168 - 24 = 144 \]

Thus, the total number of even factors of \( N \) is:

\[ \boxed{144} \]