Show that if $n$ and $k$ are positive integers with $n>1$ and all $n$ positive integers $a, a+k, ..., a+(n-1)k$ are odd primes, then $k$ is divisible by every prime less then $n$.
My solution:
$a=p*q_0+r_0$
$a+k=p*q_1+r_1$
...
$a+(n-1)k=p*q_{n-1}+r_{n-1}$
How can I apply the pigeonhole principle to my solution? The textbook solution is not similar to my solution...
