We say that a natural number is lucky if all of its digits are equal to 7. For example, 7 and 7777 are lucky, but 767 is not. João wrote on a piece of paper the first twenty lucky numbers starting from 7, and then added them up. What is the remainder of the division of this sum by 1000?
Let's observe that if a lucky number has more than 3 digits, the remainder of the division by 1000 is 777.

Thus, the remainder we are looking for is the same remainder of the division of

$$
7+77+\underbrace{777+777+\cdots+777}_{18 \text { times }}
$$

by 1000. But this number is

$$
84+18 \times 777=84+13986=14070
$$

Thus, the remainder is 70.