The length of the year on the planet Mars is exactly 697 days. If Mars has a calendar with a 12-day week, and year 0 begins on the first day of the week, what is the next year which will begin on the first day of the week?
I want to find out how many days are in a Mars year modulo 12, which means how many days are left over after dividing by 12. If I divide 697 by 12, I get a quotient of 58 and a remainder of 1. That means that every Mars year, the calendar shifts by one day of the week. For example, if year 0 begins on Monday, then year 1 begins on Tuesday, year 2 begins on Wednesday, and so on. To find the next year that begins on the same day as year 0, I need to find the smallest positive integer n such that n times 1 is a multiple of 12. That means n has to be 12, since 12 is the smallest multiple of 12. So the next year that begins on the same day as year 0 is year 12. 