Question

Prove that in each year the 13th day of some month occurs on a Friday ?

Answer 1

@ganeshie8

Answer 2

@ganeshie8

Answer 3

@BSwan

Answer 4

~~ no help ~~ \\ closed :(

Answer 5

try this
http://jwilson.coe.uga.edu/EMAT6680Su10/Newton/emat6690/Friday%20the%2013th/Fridaythe13th.html

Answer 6

sukriya :)

Answer 7

the month has to begin on Sunday for 13th to be Friday

Answer 8

next work how many months can begin on Sunday

Answer 9

0 : Sunday
1 : Monday
2 : Tuesday
...
5 : Friday
6 : Saturday

Answer 10

Actually you can show that 13th of some month between "May" and "November" is friday

Answer 11

Suppose May1st starts on day : `k`,

then June1st has to start on day : `k + 31 mod 7 = k + 3`

yes ?

Answer 12

Sorry, I was away. i was reading the pdf. The fact is that I have understood it. But How should I start the proof. I mean it is a subjective exam and it counts that.

Answer 13

@ganeshie8

my first step should be to find the month whose days start with sunday right ?

How can I write this out.

Answer 14

Okay I get it that for the 13th day of a month to be a friday, the month to start with a sunday. Then how to find the moth whose day start with monday.

Answer 15

lets start from the begin.

1. I have to find the month which start with the monday. right ? How to do this.

Answer 16

For 13th day of a month to be Friday, the first day of the month has to be Sunday.

The first day can be any one of `Sunday, Monday, Tuesday,..., Saturday`
assign `0, 1, 2,..., 6` to the days

Answer 17

Next, consider the first day of months from `May through November` :

Answer 18

why we didnt consider from january. I mean the 1st of jan can also be a sunday ????

Answer 19

we're trying to prove that there will be one month between May and November whose 13th day is Friday

Answer 20

notice that if we cound prove above, that proves the original statement

Answer 21

okay! got it. We have to prove someday is friday. okay so we are taking the month from may to november. right ?

Answer 22

yes, actually we're going to prove the starting day of one of these months is Sunday

Answer 23

which means we will take may to start with x :) then we will use modulo, but than for the next month it will be x+31(mod7) than ???

Answer 24

Exactly !
whats the value of x+31(mod 7) ?

Answer 25

x+3(mod7)

Answer 26

x+31(mod 7) = x+4*7 + 3 (mod 7) = x+3 (mod 7)

Answer 27

yes, lets create a table

Answer 28

May starts on day : x
June starts on day : x + 3

Answer 29

what about July ?

Answer 30

x+5

Answer 31

Very good

May starts on day : x
June starts on day : x + 3
July starts on day : x+5

Answer 32

calculate the first day for remaining months till November

Answer 33

okay

may : x
june:x+3
july : x+5
august : x+1
sept :x+4
oct: x+6
nov:x+2

correct ???

Answer 34

Yes !
notice that these first days arranging in some order you get :
\[\large \{x,~ x+1,~ x+2,~ x+3, ~x+4,~ x+5, ~x+6\}\]

Answer 35

Clearly they are 7 consecutive numbers, yes ?

Answer 36

yes!! :)

Answer 37

So, no matter what the value of x is,
one of them will always be divisible by 7, yes ?

Answer 38

yes which means it will be 0(mod7) so, we got the solution any day between may and nov can have a friday 13th.

Answer 39

In other words, 7 consecutive days represent a full week

Answer 40

Any 7 consecutive days represent a full week

Answer 41

ya that will represent a full week. Hmmm go it :)

Answer 42

may be a more better way to put it is :
Any 7 consecutive days contain all the 7 different days of a week (Sunday through Saturday)

Answer 43

yes, so any of the month between may and nov can have a sunday start.... :)

Answer 44

So one of the starting days in these months will be Sunday,
consequently the 13th day of that particular month would be Friday

Answer 45

Exactly !

Answer 46

Okay, I will write this problem out and show you by evening,I have coaching now. I suppose you will available in the evening. :)

Thank you bhaiya, it is very good learning with you.

Answer 47

have a nice day :)

Answer 48

thank u :)

Answer 49

Lord be with you :) Bye :)

Answer 50

hey in this question we need not to worry about leap years right? But if we had taken from jan to june than olny we need to consider the leap year fact right ?

@ganeshie8 :)

Answer 51

#typo : only

Answer 52

yes thats the reason we want to avoid dealing with feb

Answer 53

hmmmm. so that is why you have taken it from may, :)

Answer 54

yes, you could try it from march also

Answer 55

ya.. :) I have one more question can i annoy you with that too.... :)

Answer 56

lol no, sure ask :) your questions are interesting and hard !
i am about to have dinner though..

Answer 57

Q. Find the point in the closed unit disc D= {(x,y)| x^2+y^2 le 1 } at which the function f(x,y) = x+y attains its maximum.

This is a 2 variable function not included in the last course. I need it for an entrance.

Answer 58

lets work it quick by using geometry

Answer 59

f(x,y) = x+y = c
represent a set of parallel lines in xy plane, yes ?

Answer 60

okay.. lets do one thing. You have your lunch and start it tomorrow. Sounds good ? Cause I have doubts and you need to buy some time now. So, Will start fresh tomorrow ?

Answer 61

no problem , I suppose. Than I can annoy you, else it will cost your great time, consequence you shall loose your dinner :) No problem from my edge.. :)

Answer 62

its okay i have few minutes, let me finish this quick :

Notice that the maximum value for x+y corresponds to below tangent line :



why ? because beyond that, the lines x+y=c wont satisfy the points in given unit disk and the lines below that have less x and y intercepts and so the value of x+y would be less