Key Concepts and Formulas

• Function Evaluation and Parameter Solving: Given a function with unknown parameters, use provided function values to set up a system of equations and solve for the parameters.
• Sum of Divisors Formula: For a positive integer $N$ with prime factorization $N = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}$, the sum of its positive divisors is $\sigma(N) = \prod_{i=1}^{k} \frac{p_i^{a_i+1}-1}{p_i-1}$.

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Step-by-Step Solution

1. Understand the Function and Given Information We are given the function $f(x) = 2x^n + \lambda$, where $\lambda \in \mathbb{R}$ and $n \in \mathbb{N}$. We are also given that $f(4) = 133$ and $f(5) = 255$. Our goal is to find the sum of all positive integer divisors of $(f(3) - f(2))$.

2. Formulate Equations Using Given Values We substitute the given values of $x$ into the function definition:

• For $x=4$: $f(4) = 2(4^n) + \lambda = 133$ (Equation 1)
• For $x=5$: $f(5) = 2(5^n) + \lambda = 255$ (Equation 2) These equations will help us determine the unknown parameters $n$ and $\lambda$.

3. Solve for the Parameter 'n' To find $n$, we can eliminate $\lambda$ by subtracting Equation 1 from Equation 2: $(2(5^n) + \lambda) - (2(4^n) + \lambda) = 255 - 133$ $2 \cdot 5^n + \lambda - 2 \cdot 4^n - \lambda = 122$ $2(5^n - 4^n) = 122$ Divide both sides by 2: $5^n - 4^n = 61$ Now, we need to find the natural number $n$ that satisfies this equation. We can test small values of $n$:

• If $n=1$, $5^1 - 4^1 = 5 - 4 = 1 \neq 61$.
• If $n=2$, $5^2 - 4^2 = 25 - 16 = 9 \neq 61$.
• If $n=3$, $5^3 - 4^3 = 125 - 64 = 61$. Thus, $n=3$ is the solution. Since the function $g(n) = 5^n - 4^n$ is strictly increasing for $n \in \mathbb{N}$, this solution is unique.

4. Calculate the Expression $(f(3) - f(2))$ Now that we know $n=3$, we can calculate $f(3)$ and $f(2)$:

• $f(3) = 2(3^n) + \lambda$
• $f(2) = 2(2^n) + \lambda$ We are interested in the difference $(f(3) - f(2))$: $f(3) - f(2) = (2(3^n) + \lambda) - (2(2^n) + \lambda)$ $f(3) - f(2) = 2(3^n - 2^n)$ Substitute $n=3$: $f(3) - f(2) = 2(3^3 - 2^3)$ $f(3) - f(2) = 2(27 - 8)$ $f(3) - f(2) = 2(19)$ $f(3) - f(2) = 38$

5. Find the Sum of Positive Integer Divisors of 38 To find the sum of the positive integer divisors of 38, we first find its prime factorization: $38 = 2^1 \cdot 19^1$ The sum of the divisors, $\sigma(38)$, is given by the formula: $\sigma(38) = \left(\frac{2^{1+1}-1}{2-1}\right) \left(\frac{19^{1+1}-1}{19-1}\right)$ $\sigma(38) = \left(\frac{2^2-1}{1}\right) \left(\frac{19^2-1}{18}\right)$ $\sigma(38) = (4-1) \left(\frac{361-1}{18}\right)$ $\sigma(38) = 3 \left(\frac{360}{18}\right)$ $\sigma(38) = 3 (20)$ $\sigma(38) = 60$ Alternatively, we can list the divisors: 1, 2, 19, 38. Their sum is $1 + 2 + 19 + 38 = 60$.

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Common Mistakes & Tips

• Solving for $\lambda$: It is not necessary to find the value of $\lambda$ as it cancels out in the difference calculations. Focus on finding $n$ first.
• Testing values for $n$: For equations of the form $a^n - b^n = C$ where $n$ is a natural number, testing small integer values is an efficient strategy.
• Prime Factorization Accuracy: Ensure the prime factorization of the number is correct before applying the sum of divisors formula.

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Summary We were given a function $f(x) = 2x^n + \lambda$ and two points to determine the parameters. By setting up equations and solving them, we found that $n=3$. We then calculated the expression $f(3) - f(2)$ which resulted in 38. Finally, we found the sum of the positive integer divisors of 38 using its prime factorization, which yielded 60.

The final answer is $\boxed{60}$.