Suppose sum of two smallest positive divisors of an integer N is 6, while the sum of the two largest positive divisors of N is 858. What is the sum of the digits of N?

01) 13

02) 15

03) 17

04) 19

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.

Suppose sum of two smallest positive divisors of an integer N is 6, while the sum of the two largest positive divisors of N is 858. What is the sum of the digits of N?

Solve a number theory problem where the sum of the smallest divisors of an integer N is 6, and the sum of the largest divisors is 858. Learn how to use divisor properties and factorization to find N and its digit sum. Suitable for Grades 9-12.

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