There are six numbers which are not divisible by 6

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Discussion Overview

The discussion revolves around a mathematical problem involving a set of six numbers that are not divisible by 6. Participants are tasked with proving that at least two of these numbers must have a difference that is divisible by 6, exploring concepts related to the pigeonhole principle and modular arithmetic.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asserts that there are six numbers not divisible by 6 and proposes to prove that at least two of them have a difference divisible by 6.
  • Another participant references the pigeonhole principle, suggesting that since there are six possible remainders when dividing by 6 (1, 2, 3, 4, 5), at least two numbers must share a remainder, leading to a difference divisible by 6.
  • A participant elaborates on the set of six values, indicating that none can have a remainder of 0 when divided by 6, and lists the possible remainders.
  • One participant expresses confusion and requests clarification on the previous points made.
  • Another participant explains the pigeonhole principle in relation to the problem, confirming the reasoning about the limited number of remainders for numbers not divisible by 6.
  • A participant acknowledges understanding after receiving clarification.

Areas of Agreement / Disagreement

Participants generally agree on the application of the pigeonhole principle to the problem, but there is some confusion regarding the explanation and details of the argument, indicating that not all points are fully understood by everyone involved.

Contextual Notes

Some participants may not be familiar with the pigeonhole principle or modular arithmetic, which could limit their understanding of the discussion.

rajesh
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There are six numbers which are not divisible by 6.
Prove that there are atleast two numbers in this set such that the difference between them is divisible by 6.
 
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pigeon hole principle.

their are six remainders possible on division by 6. none of the numbers in your set has remainder zero so two of them must be...
 
rajesh said:
There are six numbers which are not divisible by 6.
Prove that there are atleast two numbers in this set such that the difference between them is divisible by 6.

Let's say x is our set of six values
x mod 6 <> 0 so:
x mod 6 = 1 and 2 and 3 and 4 and 5 and y

y being the sixth value.

We also know that y <> 1 <> 2 <> 3 <> 4 <> 5 <> 0
y is not contained within the range of n mod 6.

so...
 
i didnt get both of u...
please elaborate
 
do you know what the pidgeon hole prinicipal is? if you dont, it states (sort of obvious) that if you have x holes and x + 1 pidgeons and all pidgeons go to some hole, there will be one hole with at least 2 pidgeons. Applied to the problem, i think matt grime was basically show that there are only 5 possible remainders if a number is not divisible by 6, and then he just enumerated the possibilities. Is that right Matt grime?
 
Look up the pigeon hole principle and the idea of remainder or modulo arithmetic. We won't do all the work for you
 
got it...thanks
 

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