GATE CS 2015 Set 3 Q35 — Number Representations (Fixed and Floating Point)

To find the number of possible solutions for the equation $(43)_x = (y3)_8$, we first establish the constraints on the digits and bases:

• For $(43)_x$ to be a valid representation, the base $x$ must be greater than its largest digit: $x > 4$.
• For $(y3)_8$ to be a valid representation, the digit $y$ must satisfy $0 \le y < 8$.

Now, we convert both sides of the equation to base 10:

Using the constraint $x > 4$, we substitute $x = 2y$:

Combining this with the constraint $y < 8$, the possible integer values for $y$ are:

For each value of $y$, there is a corresponding unique integer value for $x = 2y$:

1. $y = 3 \implies x = 6$
2. $y = 4 \implies x = 8$
3. $y = 5 \implies x = 10$
4. $y = 6 \implies x = 12$
5. $y = 7 \implies x = 14$

Thus, there are exactly 5 possible solutions.