# There are six numbers which are not divisible by 6

1. Oct 28, 2004

### rajesh

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.

2. Oct 28, 2004

### matt grime

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...

3. Oct 28, 2004

### Alkatran

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...

4. Oct 28, 2004

### rajesh

i didnt get both of u....
plz elaborate

5. Oct 28, 2004

### T@P

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?

6. Oct 28, 2004

### 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

7. Oct 28, 2004

### rajesh

got it...thanks