Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $x_1$ and $x_2$, we have $x_1 + x_2 = -\frac{b}{a}$ and $x_1 x_2 = \frac{c}{a}$.
• Properties of Natural Numbers ($\mathbf{N}$): The set of positive integers $\{1, 2, 3, \ldots\}$.
• Divisibility Rules: Basic divisibility rules for 2 and 3. A number is divisible by 2 if it's even, and divisible by 3 if the sum of its digits is divisible by 3.

Step-by-Step Solution

Step 1: Apply Vieta's formulas to the quadratic equation

Given the quadratic equation $x^2 - 70x + \lambda = 0$, we identify the coefficients: $a=1$, $b=-70$, and $c=\lambda$. Let $\alpha$ and $\beta$ be the roots. By Vieta's formulas:

• Sum of the roots: $\alpha + \beta = -\frac{-70}{1} = 70$
  • Explanation: The sum of the roots is equal to the negative of the coefficient of the $x$ term divided by the coefficient of the $x^2$ term.
• Product of the roots: $\alpha \beta = \frac{\lambda}{1} = \lambda$
  • Explanation: The product of the roots is equal to the constant term divided by the coefficient of the $x^2$ term.

Since $\alpha, \beta \in \mathbf{N}$, both $\alpha$ and $\beta$ are positive integers.

Step 2: Analyze the conditions on $\lambda$

We have two conditions: $\frac{\lambda}{2} \notin \mathbf{N}$ and $\frac{\lambda}{3} \notin \mathbf{N}$.

• Condition 1: $\frac{\lambda}{2} \notin \mathbf{N}$ implies $\lambda$ is not divisible by 2, so $\lambda$ is odd. Since $\lambda = \alpha \beta$, both $\alpha$ and $\beta$ must be odd.
  • Explanation: An even number multiplied by any integer is even, so if $\lambda$ is odd, both its factors must be odd.
• Condition 2: $\frac{\lambda}{3} \notin \mathbf{N}$ implies $\lambda$ is not divisible by 3. Since $\lambda = \alpha \beta$, neither $\alpha$ nor $\beta$ can be divisible by 3.
  • Explanation: If either $\alpha$ or $\beta$ were divisible by 3, their product would also be divisible by 3.

Therefore, we need to find odd integers $\alpha$ and $\beta$ such that $\alpha + \beta = 70$, neither $\alpha$ nor $\beta$ is divisible by 3, and we want to minimize $\lambda = \alpha \beta$.

Step 3: Find the minimum value of $\lambda$

To minimize $\lambda = \alpha \beta$ with a fixed sum $\alpha + \beta = 70$, we want $\alpha$ and $\beta$ to be as far apart as possible. Start testing small odd numbers not divisible by 3:

• If $\alpha = 1$, then $\beta = 70 - 1 = 69$. But 69 is divisible by 3, so this is not valid.
• If $\alpha = 5$, then $\beta = 70 - 5 = 65$. Both 5 and 65 are odd and not divisible by 3.
  • Explanation: 5 is clearly not divisible by 3. The sum of the digits of 65 is 11, which is not divisible by 3, so 65 is not divisible by 3.
  • Then $\lambda = \alpha \beta = 5 \times 65 = 325$. Since 325 is odd and not divisible by 3 ($3+2+5=10$ is not divisible by 3), this is a valid value for $\lambda$.

Thus, the minimum possible value of $\lambda$ is 325.

Step 4: Evaluate the given expression

We have $\alpha = 5$, $\beta = 65$, and $\lambda = 325$. Substitute into the expression:

$\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|} = \frac{(\sqrt{5-1}+\sqrt{65-1})(325+35)}{|5-65|}$

$= \frac{(\sqrt{4}+\sqrt{64})(360)}{|-60|} = \frac{(2+8)(360)}{60} = \frac{(10)(360)}{60} = \frac{3600}{60} = 60$

Common Mistakes & Tips

• Remember that $\frac{\lambda}{k} \notin \mathbf{N}$ means $\lambda$ is not divisible by $k$.
• For a fixed sum of two numbers, their product is minimized when they are as far apart as possible.
• Double-check divisibility rules and arithmetic calculations.

Summary

We used Vieta's formulas and divisibility rules to find the minimum possible value of $\lambda$ satisfying the given conditions. We found that $\lambda = 325$, with roots $\alpha = 5$ and $\beta = 65$. Substituting these values into the given expression, we found its value to be 60.

The final answer is \boxed{60}.