Question 18

Given $(123)_5 = (x8)_y$.

1. Convert to Decimal:

   • Left-hand side in base 5:
     $$(123)_5 = 1 \times 5^2 + 2 \times 5^1 + 3 \times 5^0 = 25 + 10 + 3 = 38.$$
   • Right-hand side in base $y$:
     $$(x8)_y = x \cdot y + 8.$$

2. Set up the Equation:

   $$x \cdot y + 8 = 38 \implies x \cdot y = 30.$$

3. Apply Base and Digit Constraints:

   • In base $y$, the digit $8$ appears, so the base must be strictly greater than $8$:
     $$y > 8.$$
   • $x$ is the leading digit, so $1 \le x < y$.
   • $x$ and $y$ must be positive integers.

4. Factor Pairs of 30 with $y > 8$:

   • If $y = 30$: $x = 1$ ($1 < 30$, valid).
   • If $y = 15$: $x = 2$ ($2 < 15$, valid).
   • If $y = 10$: $x = 3$ ($3 < 10$, valid).
   • If $y = 6$: violates $y > 8$.

5. There are exactly 3 valid solutions: $(x, y) \in \{(1, 30), (2, 15), (3, 10)\}$.