Total number of divisors of `n = 3^5. 5^7. 7^9` that are in the form of `4lambda + 1; lamda >=0` is equal to

Options

A. 15
B. 30
C. 120
D. 240

Answer

To find the total number of divisors of \( n = 3^5 \cdot 5^7 \cdot 7^9 \) that are in the form of \( 4\lambda + 1 \) where \( \lambda \geq 0 \), we will follow these steps:

### Step 1: Understand the Form of Divisors
Divisors of \( n \) can be expressed in the form \( 3^a \cdot 5^b \cdot 7^c \) where \( 0 \leq a \leq 5 \), \( 0 \leq b \leq 7 \), and \( 0 \leq c \leq 9 \).

### Step 2: Identify Conditions for Divisors of the Form \( 4\lambda + 1 \)
For a number to be of the form \( 4\lambda + 1 \), it must be congruent to \( 1 \mod 4 \). We will analyze the contributions of each prime factor modulo \( 4 \):

- \( 3 \equiv 3 \mod 4 \) (odd powers will contribute \( 3 \), even powers will contribute \( 1 \))
- \( 5 \equiv 1 \mod 4 \) (any power will contribute \( 1 \))
- \( 7 \equiv 3 \mod 4 \) (odd powers will contribute \( 3 \), even powers will contribute \( 1 \))

### Step 3: Determine Valid Combinations of Exponents
To ensure the product \( 3^a \cdot 5^b \cdot 7^c \equiv 1 \mod 4 \), we need to consider the parity of \( a \) and \( c \):

- If \( a \) is even, \( 3^a \equiv 1 \mod 4 \).
- If \( a \) is odd, \( 3^a \equiv 3 \mod 4 \).
- If \( c \) is even, \( 7^c \equiv 1 \mod 4 \).
- If \( c \) is odd, \( 7^c \equiv 3 \mod 4 \).

The combinations that yield \( 1 \mod 4 \) can be summarized as:
1. \( a \) even and \( c \) even
2. \( a \) odd and \( c \) odd

### Step 4: Count Valid Combinations
- **For \( a \)** (even): Possible values are \( 0, 2, 4 \) (3 options).
- **For \( a \)** (odd): Possible values are \( 1, 3, 5 \) (3 options).
- **For \( b \)**: Since \( 5^b \equiv 1 \mod 4 \), \( b \) can take any value from \( 0 \) to \( 7 \) (8 options).
- **For \( c \)** (even): Possible values are \( 0, 2, 4, 6, 8 \) (5 options).
- **For \( c \)** (odd): Possible values are \( 1, 3, 5, 7, 9 \) (5 options).

### Step 5: Calculate Total Divisors
1. **Case 1**: \( a \) even and \( c \) even:
\[ \text{Total} = 3 \text{ (for } a) \times 8 \text{ (for } b) \times 5 \text{ (for } c) = 3 \times 8 \times 5 = 120 \]

2. **Case 2**: \( a \) odd and \( c \) odd:
\[ \text{Total} = 3 \text{ (for } a) \times 8 \text{ (for } b) \times 5 \text{ (for } c) = 3 \times 8 \times 5 = 120 \]

### Step 6: Add Both Cases
\[ \text{Total divisors of the form } 4\lambda + 1 = 120 + 120 = 240 \]

### Final Answer
The total number of divisors of \( n \) that are in the form of \( 4\lambda + 1 \) is \( \boxed{240} \).