# Divisibility Problem

#### qamaz

##### New member
$$\text{ Let } n∈N \text{ and } (a1,a2,…,a_{n})∈\mathbb{Z}^{n}. \text{ Prove that always exist } i,j∈ \underline{n} \text{ with } i≤j \text{ so } \sum\limits_{k=i}^{\\j} a_{k} \text{ divisible by n} .$$

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#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Hi, and welcome to the forum!

Hint: Consider $$\displaystyle b_j=\sum_{k=1}^ja_k$$, $j=1,\ldots,n$ and apply the pigeonhole principle to remainders when $b_j$ are divided by $n$.

#### qamaz

##### New member
What is the meaning of $$\mathbb{Z}^{n}$$?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
$$\displaystyle \mathbb{Z}$$ denotes the set of integers. If $A$ and $B$ are sets, $A\times B$ denotes the set of all ordered pairs, where the first element comes from $A$ and the second one comes from $B$. More generally, $A_1\times \dots\times A_n$ denotes the set of all $n$-tuples, i.e., of ordered sequences of length $n$, where the $i$th element comes from $A_i$ for $i=1,\ldots,n$. Finally, $A^n=A\times\dots\times A$ ($n$ times). Thus, $$\displaystyle (a_1,\ldots,a_n)\in\mathbb{Z}^n$$ means that all $a_i$ are integers.