7. How many zeros does the number

$$
(1 \underbrace{000 \ldots 001}_{2017 \text { zeros }})^{2017}-1 ?
$$

end with?
Answer 2018

Solution: Factorize:

$(1 \underbrace{000 \ldots 001}_{2017 \text { zeros }})^{2017}-1=(1 \underbrace{000 \ldots 00}_{2017 \text { zeros }} 1-1) \times(1 \underbrace{000 \ldots 00}_{2017 \text { zeros }} 1^{2016}+1 \underbrace{000 \ldots 00}_{2017 \text { zeros }} 1^{2015}+$

$\ldots .+1 \underbrace{000 \ldots 00}_{2017 \text { zeros }} 1+1$ ). The first bracket ends with 2018 zeros, while the second is not divisible by 10.