The year 1999 contains 365 days and begins on Friday. Use deductive reasoning to figure out what day of the week the year 2000 begins on. Show your method. The year 2000 is a leap year; 366 days. What day of the week does 2001 begin on? Show method. How many different calanders are possible? Show method. I am confused. Please help.
This site shows the 1999 calendar. You can also get calendars for other years at this site. On which day does each year begin? Can you figure out what the pattern is?
http://www.timeanddate.com/calendar/index.html?year=1999&country=1
There are 7 days in a week, so it's the same day of the week 7, 14, 21, 28,.. days from now. E.g., to find out what day of the week it is 30 days from now, you can use that 28 days from now it's the same day, so it will be two days later in the week (e.g. Friday--->Sunday).
So, as far as days in the week are concerned, 30 = 2, in mathematics we write:
30 Mod 7 = 2
(30 Modulo 7 = 2), which means that 30 equals 2 up to a multiple of 7.
If
X Mod 7 = A
and
Y Mod 7 = B
then
X*Y Mod 7 = A*B Mod 7
Proof:
X Mod 7 = A --->
X = A + 7 n for some integer n
Y Mod 7 = A --->
Y = B + 7 m for some integer m
Therefore:
X*Y = (A + 7 n)*(B + 7 m) = A*B + some multiple of 7
We also have that (X+Y)Mod 7 = (A + B) Mod 7, of course.
To compute 365 Mod 7, you can use that:
7 Mod 7 = 0 --->
49 Mod 7 = 7^2 Mod 7 = 0^2 = 0 --->
50 Mod 7 = 1 --->
350 Mod 7 = 7*50 Mod 7 = 0*1 = 0 ---->
365 Mod 7 = (15 + 350)Mod 7 = 15 Mod 7 = 1
So, we find that 2000 begins on Saturday and that 2001 begins on Monday.
1999 starts on Friday, so the first week ends on Thursday. 7 days in a week x 52 weeks in a year = 364 days + 1 + 365 so the year 2000 starts on a Saturday.
2000 starts on Saturday, so the first week ends on Friday. 7 x 52 = 364 + 2 (for leap year) = 366 so the year 2001 starts on Monday
Is this correct?
Now how do reason out how many calendars are possible? Is it 7, because it's possible for the year to start on each day of the week?
Thank you for the help!
To find the day of the week for the beginning of the year 2000, we'll start with the given information that the year 1999 begins on a Friday. We know that 1999 has 365 days, which is equivalent to one non-leap year. Each non-leap year advances the day of the week by one day, so we can determine the day of the week for the end of the year 1999 by adding 365 days to the starting day.
Since there are 365 days in a year and 7 days in a week, we find that 365 divided by 7 leaves a remainder of 1. This tells us that the starting day of the year 2000 will be one day ahead of the starting day of 1999. So, if 1999 starts on a Friday, the year 2000 will start on a Saturday.
Now, let's move on to the year 2001. We know that the year 2000 is a leap year, which means it has 366 days instead of the usual 365. A leap year occurs every four years and adds an extra day, February 29th. This extra day does not advance the day of the week by one, but by two days. Therefore, to determine the starting day of the year 2001, we need to account for the additional day in the leap year.
Since 366 divided by 7 leaves a remainder of 2, we know that the starting day of the year 2001 will be two days ahead of the starting day of the leap year 2000. If the year 2000 starts on a Saturday, adding two days would mean that the year 2001 would start on a Monday.
Now, let's consider the number of different calendars that are possible. In a non-leap year, there are seven possible starting days, since there are seven days in a week. However, in a leap year, the starting day can be different depending on whether it is a regular year or a leap year.
In a regular year, there are seven possible starting days. But in a leap year, there can be an additional starting day if the first day of the year falls on a Saturday. This is because the additional day in February alters the day progression. Specifically, it causes the starting day to skip one day compared to the starting day of a regular year.
So, the total number of different calendars possible is 7 for regular years and 8 for leap years when the first day of the year is a Saturday.