We've been given in class of Discrete mathematics a problem which we have to prove using the Pigeonhole principle.
I've been at it for quite a while. Problem is, English is not my first language and writing mathematics in English is even more troublesome so trying to explain what i did so far is a bit problematic.
The sentence which we have to prove:
let there be a positive integer $n$. $a_{1},a_{2},a_{3},...,a_{n}$ is an arithmetic progression in which all of it's members are co-primed to $n$ $(gcd(n,a_{n})=1)$. Show that the difference of the arithmetic progression (d) is not co-primed with $n$.
I took $a_{n+1}$ members of the arithmetic progression and put it in n+1 dovecotes. each dovecote is the amount of d's used in every member of the series. for example $a_{1}$ had 0 d's in it and $a_{n}$ had n d's in it.
each dovecot of $d$ I distributed into $n-1$ remainders of n. using the pigeonhole principles, dividing $n+1$ pigeons (which are the members of the arithmetic series) into n dovecotes, meant there were two pigeons inside the same dovecote. $a_{n+1}$ and $a_{n}$ where in the dovecot of remainder 0 (Which means $n$ divides it perfectly).
I subtracted one from the other which gave me:
$a_{n+1}-a_{1} = a_{1} + (n+1-1)d - a_{1} + (1-1)d = nd$
that is a far as I got. Sorry for the messy English! Hope I managed to explain what I did so far. I thought perhaps I should have used the euclidean algorithm at some point.