If (1235)ₓ = (3033)${}_y$, where x and y indicate the bases of the corresponding numbers, then

Number Base System Conversion Explained

This problem involves converting numbers from different bases to a common base (usually base 10, the decimal system) to solve an equation. Understanding how numbers are represented in various base systems is crucial for solving such problems.

Understanding Number Bases

In a positional number system with base B, a number $(D_n{D}_{n-1}\dots D_1{D}_0{)}_B$ is equivalent to:

$$D_n\cdot B^n+D_{n-1}\cdot B^{n-1}+\cdots +D_1\cdot B^1+D_0\cdot B^0$$

Here, $D_i$ represents a digit in the number, and $B$ is the base. For a number to be valid in base $B$, all its digits $D_i$ must be less than $B$.

- For $(1235{)}_x$, the digits are 1, 2, 3, 5. This implies that the base $x$ must be greater than the largest digit, so $x>5$.
- For $(3033{)}_y$, the digits are 3, 0, 3, 3. This implies that the base $y$ must be greater than the largest digit, so $y>3$.

Converting Numbers to Base 10

The given equation is $(1235{)}_x=(3033{)}_y$.

Let's convert both numbers to base 10:

1. **Convert $(1235{)}_x$ to base 10:** $$(1235{)}_x=1\cdot x^3+2\cdot x^2+3\cdot x^1+5\cdot x^0$$ $$=x^3+2x^2+3x+5$$
2. **Convert $(3033{)}_y$ to base 10:** $$(3033{)}_y=3\cdot y^3+0\cdot y^2+3\cdot y^1+3\cdot y^0$$ $$=3y^3+0+3y+3$$ $$=3y^3+3y+3$$

Formulating the Equation

Now, we set the base 10 equivalents equal to each other:

$$x^3+2x^2+3x+5=3y^3+3y+3$$

Solving by Testing Options

We will now test each given option by substituting the values of $x$ and $y$ into the equation and checking if the left-hand side (LHS) equals the right-hand side (RHS).From the table above, we can see that only when x = 8 and y = 6 does the left-hand side of the equation equal the right-hand side. Both $x=8$ ($8>5$) and $y=6$ ($6>3$) satisfy the base validity conditions.

Final Conclusion

The values $x=8$ and $y=6$ satisfy the given equation $(1235{)}_x=(3033{)}_y$. This means the correct option is the one stating $x=8$ and $y=6$.