Find all the positive integers $N$ with an even number of digits with the property that if we multiply the two  numbers formed by cutting the number in the middle we get a number that is a divisor of $N$ ( for example $12$ works because $1 \cdot 2$ divides $12$)
1. Let \( N = a \cdot 10^n + b \) where \( a \) has exactly \( n \) digits and \( b \) has at most \( n \) digits.
2. We need \( a \cdot 10^n + b = k \cdot a \cdot b \) for some integer \( k \).
3. Rearrange to get \( b = m \cdot a \) and \( 10^n + m = k \cdot m \cdot a \).
4. Since \( m \) must divide \( 10^n \) and \( m < 10 \) (because \( a \) and \( m \cdot a \) both have \( n \) digits), the possible values for \( m \) are \( \{1, 2, 4, 5, 8\} \).

### Case 1: \( m = 1 \)
1. The equation becomes \( 10^n + 1 = k \cdot a \).
2. Thus, \( a = b = \frac{10^n + 1}{k} \).
3. For \( a \) and \( b \) to have \( n \) digits, \( k \) must be 7.
4. Therefore, \( n \equiv 3 \pmod{6} \).
5. The solutions are \( \boxed{\text{S1 : } N = \frac{(1000^{2n-1} + 1)^2}{7} \text{ for all } n \in \mathbb{Z}_{>0}} \).

### Case 2: \( m = 2 \)
1. The equation becomes \( 10^n + 2 = 2k \cdot a \).
2. Thus, \( b = 2a \) and \( a = \frac{10^n + 2}{2k} \).
3. For \( k = 2 \) and \( n = 1 \), the solution is \( \boxed{\text{S2 : } N = 12} \).
4. For \( k = 3 \), the solutions are \( \boxed{\text{S3 : } N = \frac{(10^n + 2)^2}{6} \text{ for all } n \in \mathbb{Z}_{>0}} \).

### Case 3: \( m = 4 \)
1. The equation becomes \( 10^n + 4 = 4k \cdot a \).
2. Thus, \( b = 4a \) and \( a = \frac{10^n + 4}{4k} \).
3. For \( k = 2 \) and \( n = 2 \), the solution is \( \boxed{\text{S4 : } N = 1352} \).

### Case 4: \( m = 5 \)
1. The equation becomes \( 10^n + 5 = 5k \cdot a \).
2. Thus, \( b = 5a \) and \( a = \frac{10^n + 5}{5k} \).
3. For \( k = 3 \) and \( n = 1 \), the solution is \( \boxed{\text{S5 : } N = 15} \).

### Case 5: \( m = 8 \)
1. The equation becomes \( 10^n + 8 = 8k \cdot a \).
2. Thus, \( b = 8a \) and \( a = \frac{10^n + 8}{8k} \).
3. There are no solutions in this case.