Total number of divisors of `N=2^(5)*3^(4)*5^(10)*7^(6)` that are of the form `4n+2,n ge 1`, is equal to

Options

A. 54
B. 55
C. 384
D. 385

Answer

To find the total number of divisors of \( N = 2^5 \times 3^4 \times 5^{10} \times 7^6 \) that are of the form \( 4n + 2 \) (where \( n \geq 1 \)), we can follow these steps:

### Step 1: Understand the form of the divisors
Divisors of the form \( 4n + 2 \) can be expressed as \( 2 \times \text{(odd number)} \). This means that any divisor must have exactly one factor of 2 and the remaining factors must be odd.

### Step 2: Determine the contribution of the factor of 2
Since we need exactly one factor of 2, we can take \( 2^1 \) from \( 2^5 \). The remaining powers of 2 (i.e., \( 2^0 \) to \( 2^4 \)) are not used.

### Step 3: Identify the odd factors
The odd part of \( N \) is given by \( 3^4 \times 5^{10} \times 7^6 \). We need to count the number of ways to choose the powers of these odd primes.

### Step 4: Count the choices for each odd prime
- For \( 3^4 \): The powers can be \( 0, 1, 2, 3, \) or \( 4 \). This gives us \( 4 + 1 = 5 \) choices (including \( 3^0 \)).
- For \( 5^{10} \): The powers can be \( 0, 1, 2, \ldots, 10 \). This gives us \( 10 + 1 = 11 \) choices.
- For \( 7^6 \): The powers can be \( 0, 1, 2, \ldots, 6 \). This gives us \( 6 + 1 = 7 \) choices.

### Step 5: Calculate the total number of divisors
To find the total number of divisors of the form \( 4n + 2 \), we multiply the number of choices for each prime factor:
\[ \text{Total divisors} = 1 \times 5 \times 11 \times 7 \]
Here, \( 1 \) represents the single choice for \( 2^1 \).

### Step 6: Perform the multiplication
Calculating the product:
\[ 5 \times 11 = 55 \]
\[ 55 \times 7 = 385 \]

### Final Answer
Thus, the total number of divisors of \( N \) that are of the form \( 4n + 2 \) is \( \boxed{385} \).