Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called [i]stable[/i] if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

[i]Proposed by Evan Chen[/i]
1. **Define the side lengths**:
   Let the side lengths of the triangle be \(5a\), \(80b\), and \(112c\), where \(a\), \(b\), and \(c\) are positive integers.

2. **Apply the triangle inequality**:
   For a triangle with sides \(5a\), \(80b\), and \(112c\) to be valid, the triangle inequality must hold:
   \[
   5a + 80b > 112c
   \]
   \[
   5a + 112c > 80b
   \]
   \[
   80b + 112c > 5a
   \]

3. **Simplify the inequalities**:
   Let's focus on the first two inequalities:
   \[
   5a + 80b > 112c \quad \text{(1)}
   \]
   \[
   5a + 112c > 80b \quad \text{(2)}
   \]

4. **Rearrange inequality (2)**:
   \[
   5a > 80b - 112c
   \]
   \[
   5a > 16(5b - 7c)
   \]

5. **Use Bezout's Lemma**:
   Since \(\gcd(5, 7) = 1\), by Bezout's Lemma, there exist integers \(b_0\) and \(c_0\) such that:
   \[
   5b_0 - 7c_0 = 1
   \]

6. **Determine the smallest \(a\)**:
   From the inequality \(5a > 16\), we get:
   \[
   a > \frac{16}{5} = 3.2
   \]
   Since \(a\) must be an integer, the smallest possible value for \(a\) is 4.

7. **Calculate the smallest side length**:
   The smallest side length is \(5a\), where \(a = 4\):
   \[
   5 \times 4 = 20
   \]

The final answer is \(\boxed{20}\).