## Task A-3.3.

Determine the last 2019 digits of the number $2^{2019} \cdot 5^{2018} \cdot 9^{2017}$.
## Solution.

Since

$$
2^{2019} \cdot 5^{2018} \cdot 9^{2017}=2 \cdot 9^{2017} \cdot(10)^{2018}
$$

and $2 \cdot 9^{2017}$ is not divisible by 10, we see that the decimal representation of the given number will end with 2018 zeros.

The last digit different from zero is equal to the last digit of the number 2 . $9^{2017}$.

Since

$$
9^{2017}+1=(9+1)\left(9^{2016}-9^{2015}+\cdots+1\right)
$$

the number $9^{2017}+1$ is divisible by 10.

Therefore, the last digit of the number $9^{2017}$ is 9, so the last digit of $2 \cdot 9^{2017}$ is 8.

Note: The last digit of the number $9^{2017}$ can be determined based on the fact that the remainders of the powers of $9^{n}$ when divided by 10 repeat periodically: $1,9,1,9, \ldots$ This fact does not need to be proven by students.