--------- #1459 - Modular Math to find the day of the week.
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- A FACT in algebra is that the digits of any multiple of 9 must sum to a multiple of 9.
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------------------------- 2 * 9 = 18 , ------------ 1 + 8 = 9
------------------------- 3 * 9 = 27 , ------------ 2 + 7 = 9
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- Using this fact can result in many interesting behaviors in algebra which contains numbers that are multiples of 9 and the calendar which contains multiples of 7
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------------------------ 1,234 = 1 * (1 + 999) + 2 * ( 1 + 99) + 3 * ( 1 + 9) + 4
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------------------------ Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday repeat themselves every 7 days.
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- This is called modular arithmetic or clock arithmetic in which numbers can be thought of as wrapping around in a circle. Because the days of the week wrap around with 7 the modular math can calculate any day of the week in history.
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- To illustrate modular math. Pick any number?
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------------------------------- 4
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----------------------------Triple it = 4 * 3 = 12
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--------------------------- Add 6 to the result = 6 + 12 = 18
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-------------------------- Triple that result = 3 * 18 = 54
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-------------------------- Add the digits in your answer = 5 + 4 = 9
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- You will always get the number 9, Why?
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- It is simple algebra: Pick any number?
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------------------------------- x
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----------------------------Triple it = 3 * x
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--------------------------- Add 6 to the result = 3*x + 6
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-------------------------- Triple that result = 3 * (3x + 6) = 9x + 18
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-------------------------- This is 9 * (x + 2)
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- By tripling any number twice you are guaranteed to get a multiple of 9.
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- FACT: If a number s a multiple of 9 its digits will sum to a multiple of 9
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- FACT: A number is divisible by 9 if and only if the sum of its digits is a multiple of 9.
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- FACT: A number is divisible by 3 if and only its digits add to a multiple of 3.
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- FACT: If you take any number and scramble its digits then subtract the original number for the scrambled number you will get a multiple of 9.
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---------------------------- Take the number 1,234
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------------------------------ Add up the digits and you get 10, 9 + 1 = one more than a multiple of 9
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------------------------ 1,234 = 1 * 1000 + 2 * 100 + 3 * 10 + 4
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------------------------- 1,234 = 1 * (1 + 999) + 2 * ( 1 + 99) + 3 * ( 1 +9) + 4
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- All of these terms are multiples of 9 and you are left with 1 + 2 + 3 + 4 = 10
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----------------------------- 10 = 9 + 1
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---------------------------- Therefore 1,234 is 1 greater than a multiple of 9.
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- OK, how does this modular math work with a calendar. A calendar works with a multiple of 7.
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- We have to have a starting point. The year 1900 is zero, 0. The days of the week are:
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-------------------- Sunday = 1
-------------------- Monday = 2
------------------- Tuesday = 3
------------------- Wednesday = 4
-------------------- Thursday = 5
--------------------- Friday = 6
--------------------- Saturday = 7 or 0
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- Next we need the months of the year. This can be memorized as (12^2) + (5^2) + (6^ 2) + (12^2 +2) which is put in tabular form like this:
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------January = 1 ------ April = 0 ------- July = 0 ------ October = 1
------February = 4 ------ May = 2 ------- August = 3 ---- November = 4
------March = 4 ------- June = 5 ----- September = 6 --- December = 6
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- The code for the year starts with 1900 = 0, also 2007 = 0. OK smarty what day of the week was March 19, 1961?
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------------------ 1961 = 1900 + 61
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---------------- There are 15 leap years in 61 years, ( 4* 15 = 60)
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- There are 365 days in a year so each year the day of the week advances one year. If your birthday is on Monday this year it will be on Tuesday next year. That is because as a multiple of 7, 7 * 54 = 364, 365 is one more shift is a day. Unless it is a leap year than 366 is 2 more shifts in days.
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- Ok, back to March 19, 1961
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- We had 15 leap years so we need to add 15 to the 61 , 15 + 61 = 76
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- So 76 is the year code for 1961, but since we are working in multiples of 7 we simply this by working with 6 + (7*10) and just work with the code number 6 for 1961.
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--------------- Now back to our table, March = 4
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-------------- Year + month + day = 6 + 4 + 19 = 29
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-------------- 29 = 1 + (7 * 4)
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---------------- The number for the day of the week is 1 which is Sunday.
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- March 19, 1961 was a Sunday. Check it out.
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-How about Doug’s birthday May 7, 2012
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------------------------ 2007 = 0
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-------------------------2012 = 5 + 1 leap year
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----------------------- May = 4
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------------------------ Day = 7
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----------------------- 6 + 4 + 7 = 15 = 1 + (7*2)
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--------------- 1 is left after all the multiples of 7
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----------------- 1 is a Monday.
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- Doug’s birthday is Monday, Dave, better buy him a present. No charge for the math lesson. I had good teachers. Next year we will celebrate on a Tuesday. Take the number 1,618. Scramble the digits to 8,611. Now subtract the numbers 8,611 - 1,618 = 6,993. Add these digits. 6 + 9 + 9 + 3 = 27. Add these digits. You will always get a 9. It is modular arithmetic.
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