# Divisibility and Pigeonhole principle

Given a sequence of $p$ integers $a_1, a_2, \ldots, a_p$, show that there exist consecutive terms in the sequence whose sum is divisible by $p$. That is, show that there are $i$ and $j$, with $1 \leq i \leq j \leq p$, such that $a_i + a_{i+1} + \cdots + a_j$ is divisible by $p$.

I'm having trouble with labeling which entities are the pigeons, and which are the pigeonholes. I think somewhere down the line, there has to be more different sums than $p$, but that is just a guess.

• You may find of interest a similar Pigeonhole argument: if $\rm\:n\:$ is coprime to $10\,$ then every integer with at least $\rm\:n\:$ digits $\ne 0$ has a contiguous digit subsequence that forms an integer $\ne 0$ divisible by $\rm\:n.\ \$ Commented Sep 30, 2012 at 18:29
• Commented Nov 21, 2016 at 4:14

Hint: The holes are remainders on division by $p$. Consider $\sum_{i=1}^k a_i$ for $k=1,2,3 \ldots,p$ If any are divisible by $p$ you are done. If not, You have $p$ sums with only $p-1$ values of remainder allowed.
• @user1526710: Exactly. Then if the two that match are $k_1$ and $k_2$, the sum from $k1+1 through$k_2\$ is the one you want. Commented Sep 30, 2012 at 17:47