A natural number $n$ is called [i]nice[/i] if it enjoys the following properties:
• The expression is made ​​up of $4$ decimal digits;
• the first and third digits of $n$ are equal;
• the second and fourth digits of $n$ are equal;
• the product of the digits of $n$ divides $n^2$.
Determine all nice numbers.
1. **Define the number \( n \):**
   Let \( n = \overline{abab} \), where \( a \) and \( b \) are the digits of \( n \). This means \( n \) can be expressed as:
   \[
   n = 1000a + 100b + 10a + b = 1010a + 101b
   \]

2. **Condition for divisibility:**
   The product of the digits of \( n \) must divide \( n^2 \). The digits of \( n \) are \( a, b, a, b \), so the product of the digits is \( a^2b^2 \). We need:
   \[
   a^2b^2 \mid (1010a + 101b)^2
   \]

3. **Simplify the divisibility condition:**
   \[
   (1010a + 101b)^2 = 101^2 \cdot (10a + b)^2
   \]
   Therefore, we need:
   \[
   a^2b^2 \mid 101^2 \cdot (10a + b)^2
   \]

4. **Factor out common terms:**
   Since \( 101 \) is a prime number, it does not share any common factors with \( a \) or \( b \) unless \( a \) or \( b \) are multiples of \( 101 \), which is impossible since \( a \) and \( b \) are single digits. Thus, we need:
   \[
   a^2b^2 \mid (10a + b)^2
   \]

5. **Simplify the divisibility condition further:**
   \[
   ab \mid 10a + b
   \]

6. **Analyze the divisibility condition:**
   For \( ab \mid 10a + b \), we need \( a \mid b \) and \( b \mid 10a \). Since \( a \) and \( b \) are digits (0-9), we consider the possible values:
   - \( a \mid b \) implies \( b = ka \) for some integer \( k \).
   - \( b \mid 10a \) implies \( ka \mid 10a \).

7. **Determine possible values of \( k \):**
   Since \( a \neq 0 \), we can divide both sides by \( a \):
   \[
   k \mid 10
   \]
   The divisors of 10 are 1, 2, 5, and 10. Thus, \( k \) can be 1, 2, 5, or 10. However, since \( b \) is a single digit, \( k \) can only be 1, 2, or 5.

8. **Check each case:**
   - If \( k = 1 \), then \( b = a \). The number \( n \) becomes \( \overline{aaaa} \). The product of the digits is \( a^4 \), and \( a^4 \mid (1010a + 101a)^2 = (1111a)^2 \). This is always true.
   - If \( k = 2 \), then \( b = 2a \). The number \( n \) becomes \( \overline{a2a2} \). The product of the digits is \( 4a^4 \), and \( 4a^4 \mid (1010a + 202a)^2 = (1212a)^2 \). This is always true.
   - If \( k = 5 \), then \( b = 5a \). The number \( n \) becomes \( \overline{a5a5} \). The product of the digits is \( 25a^4 \), and \( 25a^4 \mid (1010a + 505a)^2 = (1515a)^2 \). This is always true.

9. **List all possible nice numbers:**
   - For \( k = 1 \), \( a \) can be any digit from 1 to 9 (since 0 would make \( n \) not a 4-digit number). Thus, the numbers are \( 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999 \).
   - For \( k = 2 \), \( a \) can be 1 to 4 (since \( b = 2a \) must be a single digit). Thus, the numbers are \( 1212, 2424, 3636, 4848 \).
   - For \( k = 5 \), \( a \) can only be 1 (since \( b = 5a \) must be a single digit). Thus, the number is \( 1515 \).

The final answer is \( \boxed{1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 1212, 2424, 3636, 4848, 1515} \)