Consider the equation $(123)_5 = (x8)_y$ with $x$ and $y$ as unknown. The number of possible solutions is ________.

Given the equation:

Converting both sides to base 10:

Equating both sides:

Now consider the valid constraints for base $y$ and digits $x, 8$:

1. Since the digit $8$ is in base $y$, the base must satisfy $y > 8$.

2. The leading digit must satisfy $1 \le x < y$.

3. $x$ and $y$ must be positive integers.

The factors of $30$ are $1, 2, 3, 5, 6, 10, 15, 30$.

Selecting $y > 8$:

• For $y = 10$: $x = \frac{30}{10} = 3$ (valid since $1 \le 3 < 10$)

• For $y = 15$: $x = \frac{30}{15} = 2$ (valid since $1 \le 2 < 15$)

• For $y = 30$: $x = \frac{30}{30} = 1$ (valid since $1 \le 1 < 30$)

Thus, there are 3 possible pairs $(x, y) \in \{(3, 10), (2, 15), (1, 30)\}$.