GATE CS 2014 Set 2 Q8 — Number Representations (Fixed and Floating Point)

To find the number of possible solutions for the equation $(123)_5 = (x8)_y$, we first convert both sides of the equation to decimal (base 10).

Step 1: Convert the LHS to decimal
$$(123)_5 = 1 \times 5^2 + 2 \times 5^1 + 3 \times 5^0 = 25 + 10 + 3 = 38_{10}$$

Step 2: Convert the RHS to decimal
$$(x8)_y = x \cdot y^1 + 8 \cdot y^0 = xy + 8$$

Step 3: Set up the equation
$$xy + 8 = 38 \implies xy = 30$$

Step 4: Apply the constraints of positional number systems

1. Since the digit 8 is used in the base $y$ representation $(x8)_y$, the base $y$ must be strictly greater than 8: $$y \ge 9$$
2. Since $x$ is a digit in base $y$, it must satisfy: $$0 \le x < y$$

Step 5: Find the valid integer pairs $(x, y)$
We look for positive integer factors of $30$ such that $xy = 30$ and $y \ge 9$:

• If $y = 10$, then $x = 3$ (Valid, since $3 < 10$ and $10 \ge 9$)
• If $y = 15$, then $x = 2$ (Valid, since $2 < 15$ and $15 \ge 9$)
• If $y = 30$, then $x = 1$ (Valid, since $1 < 30$ and $30 \ge 9$)

Thus, there are exactly 3 possible solutions: $(3, 10)$, $(2, 15)$, and $(1, 30)$.