Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.
1. Let \( a \) denote the side length of the square \( ABCD \). Given that the square is divided into 88 smaller squares of side length 1 and one square of side length \( n \), we can write the equation for the total area:
   \[
   88 \cdot 1^2 + n^2 = a^2
   \]
   Simplifying, we get:
   \[
   88 + n^2 = a^2
   \]

2. Rearrange the equation to isolate the difference of squares:
   \[
   a^2 - n^2 = 88
   \]
   This can be factored using the difference of squares formula:
   \[
   (a - n)(a + n) = 88
   \]

3. Since \( a \) and \( n \) are integers and \( n > 1 \), we need to find pairs of factors of 88 that satisfy the conditions \( a + n > a - n \) and both factors having the same parity (both even or both odd). The factor pairs of 88 are:
   \[
   (1, 88), (2, 44), (4, 22), (8, 11)
   \]
   However, only the pairs where both factors are even are valid:
   \[
   (2, 44), (4, 22)
   \]

4. Solve for \( a \) and \( n \) using the valid pairs:
   - For the pair \((a - n, a + n) = (4, 22)\):
     \[
     a - n = 4 \quad \text{and} \quad a + n = 22
     \]
     Adding these equations:
     \[
     (a - n) + (a + n) = 4 + 22 \implies 2a = 26 \implies a = 13
     \]
     Subtracting these equations:
     \[
     (a + n) - (a - n) = 22 - 4 \implies 2n = 18 \implies n = 9
     \]
     Thus, one solution is \( (a, n) = (13, 9) \).

   - For the pair \((a - n, a + n) = (2, 44)\):
     \[
     a - n = 2 \quad \text{and} \quad a + n = 44
     \]
     Adding these equations:
     \[
     (a - n) + (a + n) = 2 + 44 \implies 2a = 46 \implies a = 23
     \]
     Subtracting these equations:
     \[
     (a + n) - (a - n) = 44 - 2 \implies 2n = 42 \implies n = 21
     \]
     Thus, another solution is \( (a, n) = (23, 21) \).

5. Therefore, the possible lengths for the sides of \( ABCD \) are:
   \[
   \boxed{13 \text{ and } 23}
   \]