11.2. In the piggy bank, there are 1000 coins of 1 ruble, 2 rubles, and 5 rubles, with a total value of 2000 rubles. How many coins of each denomination are in the piggy bank, given that the number of 1-ruble coins is a prime number.
Answer: 1-ruble coins - 3, 2-ruble coins - 996, 5-ruble coins - 1. Solution. Let $x, y, z$ be the number of coins worth 1 ruble, 2 rubles, and 5 rubles, respectively. Then $x+y+z=1000$ and $x+2 y+5 z=2000$. Subtract the first equation, multiplied by 2, from the second. We get $x=3 z$. Thus, from the condition that $x$ is a prime number, it follows that $x=3$, and therefore, $\mathrm{z}=1$. Then $y=996$.