Let `f(x)=2x^(n)+lambda,lambda in R,n in N` and `f(4)=133,f(5)=255`.Then the sum of all the positive integer divisors of `(f(3)-f(2))` is

Options

A. `59`
B. `60`
C. `61`
D. `58`

Answer

To solve the problem, we need to find the sum of all positive integer divisors of \( f(3) - f(2) \) given the function \( f(x) = 2x^n + \lambda \) and the values \( f(4) = 133 \) and \( f(5) = 255 \).

### Step-by-Step Solution:

1. **Set Up the Equations:**
From the function definition:
\[ f(4) = 2(4^n) + \lambda = 133 \quad (1) \]
\[ f(5) = 2(5^n) + \lambda = 255 \quad (2) \]

2. **Subtract the Equations:**
Subtract equation (1) from equation (2):
\[ (2(5^n) + \lambda) - (2(4^n) + \lambda) = 255 - 133 \]
This simplifies to:
\[ 2(5^n - 4^n) = 122 \]
Dividing both sides by 2:
\[ 5^n - 4^n = 61 \quad (3) \]

3. **Solve for \( n \):**
We need to find a natural number \( n \) such that \( 5^n - 4^n = 61 \).
- For \( n = 1 \):
\[ 5^1 - 4^1 = 5 - 4 = 1 \quad (\text{not a solution}) \]
- For \( n = 2 \):
\[ 5^2 - 4^2 = 25 - 16 = 9 \quad (\text{not a solution}) \]
- For \( n = 3 \):
\[ 5^3 - 4^3 = 125 - 64 = 61 \quad (\text{solution found!}) \]

4. **Find \( \lambda \):**
Substitute \( n = 3 \) back into either equation (1) or (2) to find \( \lambda \). Using equation (1):
\[ 2(4^3) + \lambda = 133 \]
\[ 2(64) + \lambda = 133 \]
\[ 128 + \lambda = 133 \]
\[ \lambda = 5 \]

5. **Calculate \( f(3) \) and \( f(2) \):**
Now we can calculate \( f(3) \) and \( f(2) \):
\[ f(3) = 2(3^3) + 5 = 2(27) + 5 = 54 + 5 = 59 \]
\[ f(2) = 2(2^3) + 5 = 2(8) + 5 = 16 + 5 = 21 \]

6. **Find \( f(3) - f(2) \):**
Now calculate:
\[ f(3) - f(2) = 59 - 21 = 38 \]

7. **Find the Sum of Positive Integer Divisors of 38:**
The positive integer divisors of 38 are \( 1, 2, 19, 38 \).
The sum of these divisors is:
\[ 1 + 2 + 19 + 38 = 60 \]

### Final Answer:
The sum of all positive integer divisors of \( f(3) - f(2) \) is \( \boxed{60} \).