5. Antea and Barbara start their new job on the same day. Antea works three days in a row, then takes one day off and continues in the same manner (3 working days, then 1 day off). Barbara works seven days in a row, then takes three days off and continues in the same manner (7 working days, then 3 days off).

How many common days off will Antea and Barbara have in the first 1000 days of work?

## Tasks worth 10 points:
5. Antea has cycles of rest and work lasting a total of 4 days, while Barbara's cycle lasts a total of 10 days.

Thus, after 20 days, their work and rest cycles repeat in the same way.

In the first 20 days from the start of their new jobs, Antea will rest on the fourth, eighth, twelfth, sixteenth, and twentieth day, counting from the start, while Barbara will rest on the eighth, ninth, tenth, eighteenth, nineteenth, and twentieth day, counting from the start.

This means that in the first 20 days, they will have two common rest days.

In the first 1000 days of work, there will be a total of 50 such 20-day cycles, which means they will have a total of $50 \cdot 2=100$ common rest days.

Note: The partial scores mentioned are intended to assist in grading students who have not fully or correctly solved the problem. If a student has solved the problem using all necessary arguments, regardless of whether they have not mentioned some of the statements for which points are awarded, they should be graded with 6 POINTS.