What is the least number $N$ of 4-digits numbers compiled from digits $1,2,3,4,5,6,7,8$ you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen $N$ numbers?
b)At least in one, but not more than in two of chosen $N$ numbers?
To solve this problem, we need to ensure that any two different digits from the set \(\{1,2,3,4,5,6,7,8\}\) appear together in at least one of the chosen 4-digit numbers. Let's break down the solution step-by-step.

### Part (a)

1. **Define the Problem:**
   We need to find the minimum number \(N\) of 4-digit numbers such that any two different digits from the set \(\{1,2,3,4,5,6,7,8\}\) appear together in at least one of the chosen numbers.

2. **Connecting Digits:**
   Call two digits *connected* if there exists a 4-digit number with both digits in this number. For each digit \(d\), we need to ensure that it is connected to all other 7 digits.

3. **Maximum Connections per Number:**
   Each 4-digit number can connect a digit \(d\) with at most 3 other digits. This is because a 4-digit number can only contain 4 digits in total, including \(d\).

4. **Minimum Numbers Required:**
   Since each digit needs to be connected to 7 other digits, and each 4-digit number can connect a digit to 3 other digits, we need at least \(\lceil \frac{7}{3} \rceil = 3\) numbers per digit to ensure all connections.

5. **Total Numbers Calculation:**
   Since there are 8 digits, and each digit needs 3 numbers to connect to all other digits, we might initially think we need \(3 \times 8 = 24\) numbers. However, each number can connect multiple digits simultaneously. Therefore, we need to find the minimum set of numbers that ensures all pairs are connected.

6. **Constructing the Set:**
   We can construct a set of 6 numbers that ensures all pairs of digits are connected:
   - 1234
   - 1256
   - 1278
   - 3478
   - 3456
   - 5678

   By checking these numbers, we can see that every pair of digits from \(\{1,2,3,4,5,6,7,8\}\) appears together in at least one of these numbers.

Thus, the minimum number \(N\) of 4-digit numbers required is 6.

The final answer is \(\boxed{6}\).