## 12. Math Puzzle $5 / 66$

A collector of tin soldiers has an almost uncountable number of figures. When arranging them in different formations, he noticed something peculiar.

When the figures are arranged in rows of 2, 3, 4, 5, 6, 7, 8, 9, and 10, there is always exactly one figure left over. Only when arranging them in rows of 11, no figure is left over.

Since he had paid attention in math class, he was able to determine the number of his figures after a short calculation. How many tin soldiers did the collector have?
From the divisors 2, 3, 4, 5, 6, 7, 8, 9, and 10, 2520 follows as the least common multiple. When the collector arranges his tin soldiers in rows of 2, 3, 4, 5, 6, 7, 8, 9, or 10, he always has one figure left over, but no figure is left over when arranging in rows of 11. This results in the equation

$$
11 x = 2520 y + 1 \rightarrow x = 229 y + \frac{y + 1}{11}
$$

If $u = \frac{y + 1}{11}$, then $y = 11 u - 1$ and $x = 2520 u - 229$. The assignments are shown in the value table:

| $u$ | $x$ | $y$ | $11 x$ | $2520 y$ |
| :---: | :---: | :---: | :---: | :---: |
| 1 | 2291 | 10 | 25201 | 25200 |
| 2 | 4811 | 21 | 52921 | 52920 |
| 3 | 7331 | 32 | 80641 | 80640 |

The column $11 x$ records the number of tin soldiers the collector has. 25201 is the smallest possible number and thus the solution.