If the first day of a year (other than the leap year) was Friday, then which one of the following was the last day of that year ?
To find the day of the week for the last day, we need to find the number of days after the first day modulo 7.
Number of days in a non-leap year = 365.
365 days = 52 weeks + 1 day.
52 weeks contain exactly 52 * 7 = 364 days.
If the first day (Day 1) is Friday, then after 52 full weeks (on Day 365 – 1 = Day 364), the day of the week will be the same as the first day, i.e., Friday.
Day 364 is a Friday.
The last day of the year is Day 365. It will be the day after Friday, which is Saturday. This is incorrect.
Let’s rethink.
Day 1 is Friday.
After 7 days (Day 8), it’s Friday again.
After 364 days (52 weeks), the day is the same as Day 1. So, Day 364 is a Friday.
Day 365 is the day after Day 364. So, Day 365 is Saturday. This is also incorrect, as the well-known property is that the first and last days are the same.
Let’s think about the “extra” day.
The days of the year are Day 1, Day 2, …, Day 365.
Day 1 = Friday.
Day 2 = Saturday.
Day 7 = Thursday.
Day 8 = Friday (1 week after Day 1).
Day (1 + 7k) is Friday.
We want to find the day for Day 365.
365 = 1 + 364.
The number of “extra” days after Day 1 is 364.
364 modulo 7 is 0 (364 = 52 * 7).
This means Day 365 is 0 days past Friday in the weekly cycle, counting from Day 1.
So, if Day 1 is Friday, Day (1 + 364) i.e., Day 365, is Friday + 0 days = Friday.
In a leap year (366 days), the last day would be one day later in the week than the first day.