Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
1. **Identify the primes less than or equal to 17:**
   The primes less than or equal to 17 are: 
   \[
   2, 3, 5, 7, 11, 13, 17
   \]
   There are 7 such primes.

2. **Consider the number 1:**
   The number 1 is a special case because it is not divisible by any prime number. Therefore, it can be included in our set without violating the condition that no two numbers share a common divisor greater than 1.

3. **Form the set:**
   We can form a set of numbers that includes the 7 primes and the number 1. This set is:
   \[
   \{1, 2, 3, 5, 7, 11, 13, 17\}
   \]

4. **Verify the condition:**
   - The number 1 does not share any common divisor greater than 1 with any other number.
   - Each prime number is only divisible by 1 and itself, so no two primes share a common divisor greater than 1.

5. **Count the numbers in the set:**
   The set contains 8 numbers.

Therefore, the largest integer \( N \) such that one can select \( N \) different positive integers, none of which is larger than 17, and no two of which share a common divisor greater than 1, is 8.

The final answer is \(\boxed{8}\)