2. Several brothers divided a certain amount of gold coins. The first brother took 100 gold coins and $\frac{1}{6}$ of the remainder. The second brother took 200 gold coins and $\frac{1}{6}$ of the new remainder, and so on. The last brother took whatever was left. It turned out that each brother took an equal number of gold coins. How many brothers participated in the division?
Solution. The first brother received $100+\frac{1}{6}(\mathrm{x}-100)$ gold coins, and the second brother received: $200+\frac{1}{6}\left(x-200-100-\frac{1}{6}(x-100)\right)$ gold coins. Since all the brothers received the same number of gold coins, we have $100+\frac{1}{6}(x-100)=200+\frac{1}{6}\left(x-200-100-\frac{1}{6}(x-100)\right)$ from which it follows that $x=2500$. Therefore, the first brother received 500 gold coins. Thus, there were $2500: 500=5$ brothers.