Let's analyze the given conditions step by step.

Step 1: Understanding the problem

• Condition 1: The sum of the two smallest positive divisors of $N$ is 6.
• Condition 2: The sum of the two largest positive divisors of $N$ is 858.
• We need to find the sum of the digits of $N$.

Step 2: Determine the smallest divisors

The smallest positive divisor of any integer is always 1. Let the second smallest divisor be $d$. Therefore: $1 + d = 6 \implies d = 5$ Thus, the two smallest divisors are 1 and 5.

Step 3: Determine the form of $N$

Since 5 is a divisor of $N$, $N$ must be a multiple of 5.

Step 4: Determine the largest divisors

The largest divisor of any number is the number itself, $N$. Let the second largest divisor be $N/d'$, where $d'$ is another divisor of $N$. According to the problem: $N + \frac{N}{d'} = 858$ Rewriting the equation: $N \left(1 + \frac{1}{d'}\right) = 858 \implies N \left(\frac{d' + 1}{d'}\right) = 858$ Thus, $N = \frac{858 \cdot d'}{d' + 1}$

Step 5: Testing divisors

To solve for $N$, we need to find suitable $d'$ such that $N$ is a multiple of 5. Testing values of $d'$:

• For $d' = 5$: $N = \frac{858 \cdot 5}{5 + 1} = \frac{4290}{6} = 715$ Thus, $N = 715$.

Step 6: Verify the conditions

1. Smallest divisors: 1 and 5 (sum is 6).
2. Largest divisors: 715 and $\frac{715}{5} = 143$ (sum is 858).

Both conditions are satisfied.

Step 7: Sum the digits of $N$

The sum of the digits of $715$ is: $7 + 1 + 5 = 13$

Answer

The sum of the digits of $N$ is: $\boxed{13}$

Correct Option:

01) 13

Would you like further details on any step, or have any questions?

Relative Questions:

1. What is the importance of divisors in number theory?
2. How can we systematically find all divisors of a number?
3. Why is 1 always considered the smallest divisor of any integer?
4. How does factoring help in solving divisor-related problems?
5. Can the problem be solved using a different approach?

Tip:

When solving problems involving divisors, always consider both small and large divisors, as their relationships often reveal useful patterns.