Question

CHALLENGE QUESTION 3

Let S be a subset of {1,2,3,...,2013} such that no two members of S differ by 4 or 7. The largest number of members of S is divisible by

(a) 2 (b) 3 (c) 5 (d) 7

Answer 1

@mukushla
@waterineyes

Answer 2

@UnkleRhaukus

Answer 3

2013/3=671?

Answer 4

yes

Answer 5

Difference should not be of 4 or 7 @UnkleRhaukus

Answer 6

huh?

Answer 7

S can be written as :

{ 11k + 1, 11k + 2, 11k + 3, 11k + 4, ..... }


k = 0, 1, 2, ... .

Answer 8

Find the multiple of 11 near 2013..

Answer 9

I think it is the multiple..

Answer 10

2006 will be the largest number in my opinion..

Answer 11

And it is divisible by 2 only..

Answer 12

If I am right then I got the subset S as:

S : {1, 2, 3, 4, 12, 13, 14, 15, 23, 24, 25, 26, 34, 35, 36, 37, 45, 46, 47, 48 ......}


Now what are the numbers that are missing in it:

5-11
16-22
27-33
38-44




Look for the last pattern:

The numbers are divisible by 11..

As 2013 is also divisible by 11 so:

The last numbers that will not include in the set are:

2007-2013


So the largest number will be 2006..

May be I am wrong also..

Answer 13

\(11k + (1 \ to\ 4) \)

k = 0 : 1, 2, 3, 4

k = 1 : 12, 13, 14, 15
.
.
.


so from (1 to 2013), for every 11 numbers, we have 4 consecutive numbers that belong to set S

so there will be total of (2013/11) * 4 = 732 elements in set S.


# numbers divisible by 2 in set S
if u have 4 consecutive numbers, only 2 numbers will be divisible by 2.
so number of elements in S divisible by 2 = 732/2 = 366

Answer 14

Question is asking about the largest number..

Answer 15

So we have to find the largest number of subset S..

Answer 16

Put k = 183..

Answer 17

oohhh lol i see it now :) thanks @waterineyes .... i interpreted it in a complicated way

Answer 18

Sorry not 183..

put k = 182

Answer 19

yes k = 0 to 182 covers entire set...

Answer 20

You will get: 2003, 2004, 2005, 2006


So 2006 will be the greatest here..

Answer 21

For k = 183, 2014, 2015 etc which are not our requirements..

So, 2006 will be the largest number..

Answer 22

@waterineyes kudos to you.......... very gud approach of problem solving.......well regarding the answer and the correct way I will tell you that later...........

Answer 23

Sure..

This is what Mathematics is..

We can solve the question in many ways..

That is what I Love about Mathematics..

Answer 24

GOOD JOB!

Answer 25

@nitz , Using properties of Factorials :

\(GOOD \; JOB! = (GOOD \;JOB) \times (GOOD \;JOB - 1)\times(GOOD \;JOB -2).... 1\)

Answer 26

Hey @vishweshshrimali5 I was just kidding..

Hope you don't mind..

Answer 27

No Problem............ I never mind such things...... Even I liked it much didn't have more medals to give otherwise it would have been yours more........

Answer 28

@waterineyes are you ready for another challenge ?

Answer 29

I can try for that challenge @vishweshshrimali5 ..