Proving a-b=9 for natural numbers in a set of 100 with a card of 55?

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Homework Help Overview

The problem involves a set of natural numbers, specifically a set E containing 55 elements where each element is between 1 and 100. The task is to prove that there exist at least two numbers a and b in this set such that the difference a - b equals 9.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to use a form of number theory involving modular arithmetic and considers various representations of numbers in relation to their remainders when divided by 9. Some participants question the clarity of the notation used and suggest alternative sets to explore the maximum cardinality of numbers that do not differ by 9.

Discussion Status

The discussion is ongoing, with participants exploring different mathematical approaches and questioning the assumptions made in the original post. Some guidance has been offered regarding the structure of sets that could potentially meet the problem's requirements, but no consensus has been reached.

Contextual Notes

There is a noted complexity in the notation used by the original poster, which has led to some confusion among participants. Additionally, the constraints of the problem, such as the specific cardinality of the set and the range of natural numbers, are under discussion.

Andrax
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Homework Statement


let E be set of natural numbers where 0<n≤100 Card E = 55
Prove that there exists atleast 2 numbers a and b in each set where a-b=9

Homework Equations



\existsk1\in[[ ]] x1=9k1+r1 r1\in[[ ]]
\existsk2\in[[ ]] x1=9k2+r2 r2\in[[ ]]
.
.
.
\existsk55\in[[ ]] x1=9k55+r55 r2\in[[ ]]
(using derkil theory usually solves these kinds of problems )
but the above didn't help me, anyway i need to make sure that i could get any numbers from 1 to 100 in every equationi tried doing this
\existsk1\in[[1, 9 ]] x1=9k1+r1 r1\in[[1,54 ]]
\existsk2\in[[1 , 9 ]] x1=9k2+r2 r2\in[[ 1,54 ]]
.
.
.
\existsk55\in[[1 ,9 ]] x1=9k55+r55 r2\in[[1,54 ]]
we have 55 r and these r are all in a setB Card b = 54
then atleast there exists ri and rj where ri = rj
so there exists xi and xj such as xi - xj = 9(ki-kj)+ri-ri
xi-xj=9(ki-kj)
now this sadly doesn't get me anywhere because ki - kj must be equal to 1 or -1 .. ( i hope you guys are getting what I'm trying to do here)
anyway i randomly noticed that 54/6 = 9 so i tried another way ..
\existsk1\in[[ [2] , [3] ]] x1=54/k1+r1 r1\in[[ ]]
\existsk2\in[[ [2] , [3] ]] x1=54/k2+r2 r2\in[[ ]]
.
.
.
\existsk55\in[[ [2] , [3] ]] x1=54/k55+r55 r2\in[[ ]]
now tis way "might" work if and only if the ki and kj are different we will have
xi-xj= 54/ki - 54/kj
=(54ki-54kj)/kikj
but the problem on this one is that i can't find a set which contains r's and it's card <55..
i really need help on this problem
 
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I don't understand your notation.

You could consider the sets {1,10,...}, {2,11,...}, ...
Can you calculate the maximal cardinality of a set where no two numbers have a difference of 9?
 
Thanks I knew I was just fooling IA used the same notation in a similar exercise
 
Andrax said:

Homework Statement


let E be set of natural numbers where 0<n≤100 Card E = 55
Prove that there exists atleast 2 numbers a and b in each set where a-b=9

Homework Equations



\existsk1\in[[ ]] x1=9k1+r1 r1\in[[ ]]
\existsk2\in[[ ]] x1=9k2+r2 r2\in[[ ]]
.
.
.
\existsk55\in[[ ]] x1=9k55+r55 r2\in[[ ]]
(using derkil theory usually solves these kinds of problems )
but the above didn't help me, anyway i need to make sure that i could get any numbers from 1 to 100 in every equationi tried doing this
\existsk1\in[[1, 9 ]] x1=9k1+r1 r1\in[[1,54 ]]
\existsk2\in[[1 , 9 ]] x1=9k2+r2 r2\in[[ 1,54 ]]
.
.
.
\existsk55\in[[1 ,9 ]] x1=9k55+r55 r2\in[[1,54 ]]
we have 55 r and these r are all in a setB Card b = 54
then atleast there exists ri and rj where ri = rj
so there exists xi and xj such as xi - xj = 9(ki-kj)+ri-ri
xi-xj=9(ki-kj)
now this sadly doesn't get me anywhere because ki - kj must be equal to 1 or -1 .. ( i hope you guys are getting what I'm trying to do here)
anyway i randomly noticed that 54/6 = 9 so i tried another way ..
\existsk1\in[[ [2] , [3] ]] x1=54/k1+r1 r1\in[[ ]]
\existsk2\in[[ [2] , [3] ]] x1=54/k2+r2 r2\in[[ ]]
.
.
.
\existsk55\in[[ [2] , [3] ]] x1=54/k55+r55 r2\in[[ ]]
now tis way "might" work if and only if the ki and kj are different we will have
xi-xj= 54/ki - 54/kj
=(54ki-54kj)/kikj
but the problem on this one is that i can't find a set which contains r's and it's card <55..
i really need help on this problem
As posted, this is pretty much gibberish.
Andrax said:
Thanks I knew I was just fooling IA used the same notation in a similar exercise
Were you asking a serious question? It's to tell from the work you show.