10.5. Currently, there are coins of 1, 2, 5, and 10 rubles. Indicate all monetary amounts that can be paid with both an even and an odd number of coins. (You can use identical coins.)
Answer: Any amount of money greater than 1 ruble can be paid with either an even or odd number of coins.

Solution. Any amount of money greater than 10 rubles can be made up of 10-ruble and 1-ruble coins. The parity of the number of coins can be changed by exchanging 10 rubles for 2 coins of 5 rubles. An amount of money less than 11 rubles can be represented using 1-ruble coins, and the parity can be easily changed by replacing 2 one-ruble coins with a 2-ruble coin. Thus, any amount of money greater than 1 ruble can be paid with either an even or odd number of coins.