Question

If \( (1235)_x = (3033)_y \), where \( x \) and \( y \) indicate the bases of the corresponding numbers, then

• A. \( x = 7 \) and \( y = 5 \)
• B. \( x = 8 \) and \( y = 6 \)
• C. \( x = 6 \) and \( y = 4 \)
• D. \( x = 9 \) and \( y = 7 \)

Hint:

When solving for bases, express both numbers in terms of their base-10 equivalents and solve the resulting equation.

Answer 1

The Correct Option is B

We are given the equation \( (1235)_x = (3033)_y \), where \( x \) and \( y \) are the unknown bases. To solve this, we need to convert both numbers into base-10 and equate them. - For \( (1235)_x \), using base \( x \), we have: \[ 1235_x = 1 \cdot x^3 + 2 \cdot x^2 + 3 \cdot x + 5. \] - For \( (3033)_y \), using base \( y \), we have: \[ 3033_y = 3 \cdot y^3 + 0 \cdot y^2 + 3 \cdot y + 3. \] Equating the two expressions: \[ 1 \cdot x^3 + 2 \cdot x^2 + 3 \cdot x + 5 = 3 \cdot y^3 + 0 \cdot y^2 + 3 \cdot y + 3. \] After solving, we find that \( x = 8 \) and \( y = 6 \) satisfies the equation. Therefore, the correct answer is (B). Final Answer: \( x = 8 \) and \( y = 6 \)