To solve this, let’s break down the conditions:

1. Let $x$ and $y$ be the two digits of a two-digit number $xy$ (where $x$ is the tens place and $y$ is the units place).
2. The number $xy$ can be expressed as $10 \times x + y$, as each two-digit number is structured with a tens digit and a units digit.
3. We want to find integers $a$ and $b$ such that: $a \times x + b \times y = 10 \times x + y$ for all two-digit numbers from 01 to 99.

Now, I’ll generate a table where, for each two-digit number $xy$, there are values of $a$ and $b$ that satisfy this equation. Let's calculate this and print the table.Here is the table of results for each two-digit number $xy$ from 01 to 99, listing values of $x$, $y$, and valid multipliers $a$ and $b$ that satisfy the equation $a \times x + b \times y = 10 \times x + y$:

\hline \text{Number (xy)} & x & y & a & b & \text{Result} \\ \hline 01 & 0 & 1 & 1 & 1 & 1 \\ 01 & 0 & 1 & 2 & 1 & 1 \\ 01 & 0 & 1 & 3 & 1 & 1 \\ ... & ... & ... & ... & ... & ... \\ 99 & 9 & 9 & 6 & 5 & 99 \\ 99 & 9 & 9 & 7 & 4 & 99 \\ 99 & 9 & 9 & 8 & 3 & 99 \\ 99 & 9 & 9 & 9 & 2 & 99 \\ 99 & 9 & 9 & 10 & 1 & 99 \\ \hline \end{array}$$ Would you like more details or have any specific questions? ### Follow-Up Questions: 1. How does the table change if we extend $$a$$ and $$b$$ beyond 10? 2. Are there unique values of $$a$$ and $$b$$ for each number, or are there patterns? 3. Can we find an upper limit for $$a$$ or $$b$$ in this context? 4. What if the number range extends beyond 99? 5. How do patterns in values of $$a$$ and $$b$$ relate to properties of $$x$$ and $$y$$? ### Tip Always check each constraint carefully in equations with multiple variables to ensure clarity and correctness of the solution.