## Task 5 - 250735

In a study group, Rainer presents his classmates with the following task: He takes three cards, each with a two-digit number, and holds them in his hand so that no one else can see the numbers, and the six digits of the three cards read as a six-digit number when placed side by side.

He does this (with the same cards) several times until he has considered every possible order of the three cards exactly once. Each time, he notes down the six-digit number (also in a way that only he can see his notes). Then he adds up all the noted numbers. He then tells his classmates:

"One card has the number 15, another has the number 23. The third card has a two-digit number that I will not reveal. The sum of all the noted numbers is 1373736."

Determine whether there is exactly one two-digit number that can be on the third card under the given conditions according to Rainer's statements! If this is the case, determine this two-digit number!
If the sought two-digit number is initially denoted by a box $\square$, then there are exactly the following possibilities for the sequence of the three cards:

| 15 | 23 | $\square$ |
| :---: | :---: | :---: |
| 15 | $\square$ | 23 |
| 23 | 15 | $\square$ |
| 23 | $\square$ | 15 |
| $\square$ | 15 | 23 |
| $\square$ | 23 | 15 |

Without considering all the digits in the boxes, the sum is

$$
\begin{array}{lll}
76 & 76 & 76
\end{array}
$$

According to Rainer's statement, with the digits in the boxes considered, the sum is

$$
\begin{array}{lll}
137 & 37 & 76
\end{array}
$$

Therefore, Rainer's statements are satisfied if and only if an addition of the following form

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0489.jpg?height=103&width=278&top_left_y=1799&top_left_x=889)

results in a value that, when added to 767676, gives the sum 1373736. Since 1373736 - 767676 = 606060, this is true if and only if the addition results in 606060. This is the case if the addition $\square + \square$ results in the sum 60, which is true if and only if the number on the third card is 30.

2. Alternative solution:

Let $a$ and $b$ (in that order) be the sought two digits. Then Rainer adds the numbers

$$
\begin{array}{rcr}
152300 + 10a + b & ; & 150023 + 100 \cdot (10a + b) \\
231500 + 10a + b & ; & 230015 + 100 \cdot (10a + b) \\
1523 + 10000 \cdot (10a + b) & ; & 2315 + 10000 \cdot (10a + b)
\end{array}
$$

Their sum is $767676 + 20202 \cdot (10a + b)$. Therefore, Rainer's statements are satisfied if and only if $767676 + 20202 \cdot (10a + b) = 1373736$. This is equivalent to $20202 \cdot (10a + b) = 606060$, and $(10a + b) = 30$.

Thus, Rainer's statements are satisfied if and only if the number on the third card is 30.