Problem:
Al and Barb start their new jobs on the same day. Al's schedule is workdays followed by rest-day. Barb's schedule is work-days followed by rest-days. On how many of their first days do both have rest-days on the same day?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Al follows a -day cycle, and Barb follows a -day cycle. The least common multiple of and is . Together they follow a -day cycle, and there are such cycles in days. Let us number the days in each cycle as . Al rests on days and . Barb rests on days and . They both rest on days and , which is days in each cycle. Thus they have common rest-days during their first days.
Make a table:
Observe that this pattern of days repeats. Since there are such cycles in days, it follows that the number of common rest-days is .