GATE CS 2004 Q19 — Number Representations (Fixed and Floating Point)

To find the possible values of $x$ and $y$, we first convert both numbers from their respective bases to base-10:

Since $(73)_x = (54)_y$, we equate the two expressions:

Additionally, we must satisfy the base constraints for the digits present in each number:

• For $(73)_x$ to be valid, the base $x$ must be strictly greater than the largest digit in the number, so $x > 7$.
• For $(54)_y$ to be valid, the base $y$ must be strictly greater than the largest digit in the number, so $y > 5$.

Now, let's test the given options against the equation $7x - 5y = 1$:

• Option A: $x = 8, y = 16$
  $7(8) - 5(16) = 56 - 80 = -24 \neq 1$
• Option B: $x = 10, y = 12$
  $7(10) - 5(12) = 70 - 60 = 10 \neq 1$
• Option C: $x = 9, y = 13$
  $7(9) - 5(13) = 63 - 65 = -2 \neq 1$
• Option D: $x = 8, y = 11$
  $7(8) - 5(11) = 56 - 55 = 1$ (This satisfies the equation, and both $x > 7$ and $y > 5$ hold true).

Thus, the correct option is D.