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

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The discussion revolves around proving that in a set of 55 natural numbers from 1 to 100, there exist at least two numbers a and b such that a - b = 9. The initial approach involves using modular arithmetic and the concept of residues, but the user struggles to find a suitable set of residues that meets the criteria. They also explore different methods, including dividing 54 by potential values, but encounter difficulties in ensuring the cardinality remains below 55. Suggestions are made to consider sets with specific differences, like {1,10,...}, to calculate the maximum cardinality without two numbers differing by 9. The conversation highlights the complexity of the problem and the need for clearer notation and reasoning.
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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 dosen'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 dosen'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.
 

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