Let $n$ be a two-digit number represented as $n = 10x + y$, where $x$ is the tens digit and $y$ is the units digit.

The problem is to find which multiple of $y$ and which multiple of $x$ can sum up to $n$. Essentially, we want to express $n$ as:

$n = a \cdot y + b \cdot x$

where $a$ and $b$ are integers.

Since $n$ ranges from $01$ to $99$, we will explore how to achieve this decomposition for each $n$.

Let's try to break this down with a few examples:

1. Example: $n = 12$

   • $x = 1$, $y = 2$
   • We want to find $a$ and $b$ such that $12 = a \cdot 2 + b \cdot 1$.
   • Possible solution: $a = 5$, $b = 2$:
     • $5 \cdot 2 + 2 \cdot 1 = 10 + 2 = 12$.

2. Example: $n = 23$

   • $x = 2$, $y = 3$
   • We want to find $a$ and $b$ such that $23 = a \cdot 3 + b \cdot 2$.
   • Possible solution: $a = 5$, $b = 4$:
     • $5 \cdot 3 + 4 \cdot 2 = 15 + 8 = 23$.

3. Example: $n = 04$

   • $x = 0$, $y = 4$
   • We want to find $a$ and $b$ such that $04 = a \cdot 4 + b \cdot 0$.
   • Possible solution: $a = 1$, $b = 0$:
     • $1 \cdot 4 + 0 \cdot 0 = 4$.

4. Example: $n = 57$

   • $x = 5$, $y = 7$
   • We want to find $a$ and $b$ such that $57 = a \cdot 7 + b \cdot 5$.
   • Possible solution: $a = 4$, $b = 5$:
     • $4 \cdot 7 + 5 \cdot 5 = 28 + 25 = 57$.

5. Example: $n = 81$

   • $x = 8$, $y = 1$
   • We want to find $a$ and $b$ such that $81 = a \cdot 1 + b \cdot 8$.
   • Possible solution: $a = 1$, $b = 10$:
     • $1 \cdot 1 + 10 \cdot 8 = 1 + 80 = 81$.

General Approach

1. Decompose $n$ into $x$ and $y$.
2. Determine values $a$ and $b$ such that $n = a \cdot y + b \cdot x$.
3. Verify the solution.

Would you like to see the calculations for more specific values of $n$, or any other detailed explanation?

Relative Questions:

1. How can we find $a$ and $b$ systematically for any $n$ between 01 and 99?
2. What are the constraints on $a$ and $b$ for a valid solution?
3. Is it possible to have more than one solution for the same $n$?
4. What happens if $x$ or $y$ equals zero—how does that affect the multiples?
5. How can we generalize the solution for $n$ with more than two digits?

Tip:

For small values of $n$, trying different values of $a$ and $b$ manually can help quickly verify solutions. For larger ranges, programming methods or systematic approaches like Diophantine equations can be useful.