564 CHAPTER 9 Mathematical Systems Whenever a process is repetitive, modular arithmetic may be helpful in answering some questions about the process. Now let’s look at an application of modular arithmetic. Example 5 Work Schedule Jackson works as a firefighter. His working schedule is to work for 6 days, and then he gets 2 days off. If today is the third day that Jackson has been working, determine the following. a) Will he be working 60 days from today? b) Will he be working 82 days from today? c) Was he working 124 days ago? Solution a) Since Jackson works for 6 days and then gets 2 days off, his working schedule may be considered a modular 8 system. That is, 8, 16, 24, … days from today will be just like today, the third day of the 8-day cycle. If we divide 60 by 8, we obtain ) 7 8 60 56 4 remainder ← Therefore, in 60 days Jackson will go through 7 complete cycles and will be 4 days further into the next cycle. If we let W represent a working day and N represent a nonworking day, then Jackson’s cycle may be represented as follows: W W W W W W N N today 4 days from today ↑ ↑ Notice that 4 days from today will be his first nonworking day of this cycle. Therefore, Jackson will not be working 60 days from today. b) We work this part in the same way we worked part (a). Divide 82 by 8 and determine the remainder. ) 10 8 82 80 2 remainder ← Thus, in 82 days it will be 2 days later in the cycle than it is today. Because he is currently in day 3 of his cycle, Jackson will be in day 5 of his cycle and will be working. c) This part is worked in the same way as parts (a) and (b), but once we determine the remainder we must move backward in the cycle. ) 15 8 124 120 4 remainder ← Thus, 124 days ago is equivalent to 4 days earlier in the cycle. Marking day 3 of the cycle (indicated by the word today) and then moving 4 days backwards Quadxeon/Shutterstock
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