To solve the problem, we need to determine how many positive integers \( N \) give a remainder of 8 when 2008 is divided by \( N \), with the condition that \( N > 8 \).
### Step-by-Step Solution:
1. **Understanding the Remainder Condition**:
When 2008 is divided by \( N \) and gives a remainder of 8, it can be expressed mathematically as:
\[
2008 \equiv 8 \, (\text{mod} \, N)
\]
This implies that:
\[
2008 - 8 = 2000 \text{ is divisible by } N
\]
Therefore, \( N \) must be a divisor of 2000.
2. **Finding the Prime Factorization of 2000**:
We can find the prime factorization of 2000:
\[
2000 = 2^4 \times 5^3
\]
3. **Calculating the Number of Divisors**:
To find the total number of positive divisors of a number from its prime factorization, we use the formula:
\[
\text{Number of divisors} = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1)
\]
where \( e_i \) are the powers of the prime factors. For 2000:
\[
(4 + 1)(3 + 1) = 5 \times 4 = 20
\]
So, 2000 has 20 positive divisors.
4. **Identifying Divisors Greater than 8**:
Now, we need to find which of these divisors are greater than 8. First, we will list all the divisors of 2000:
- The divisors of 2000 are: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 250, 400, 500, 1000, 2000.
5. **Counting Divisors Greater than 8**:
From the list of divisors, we can see:
- Divisors less than or equal to 8: 1, 2, 4, 5, 8 (total of 5 divisors).
- Divisors greater than 8: 10, 16, 20, 25, 40, 50, 80, 100, 200, 250, 400, 500, 1000, 2000 (total of 15 divisors).
6. **Final Count**:
Therefore, the number of positive integers \( N \) that give a remainder of 8 when 2008 is divided by \( N \) and where \( N > 8 \) is:
\[
\text{Total divisors} - \text{Divisors } \leq 8 = 20 - 5 = 15
\]
### Conclusion:
The answer is \( \boxed{15} \).
