The problem you uploaded states the following:

Let $a_1, a_2, \dots, a_n$ be a sequence of integers. Show that there exist integers $j$ and $k$ with $1 \leq j \leq k \leq n$ such that the sum $\sum_{i=j}^k a_i$ is a multiple of $n$.

Solution Outline:

This is a classic problem in number theory, often solved using the pigeonhole principle.

Step-by-Step Solution:

1. Partial Sums: Define partial sums $S_k$ as: $S_k = \sum_{i=1}^k a_i \quad \text{for} \quad k = 1, 2, \dots, n$ These partial sums represent the sum of the first $k$ elements of the sequence. Now consider the sum $S_k - S_j$ for $j < k$, which is the sum of elements from $a_{j+1}$ to $a_k$.

2. Modulo $n$: Since we are interested in finding a sum that is a multiple of $n$, we look at these sums modulo $n$. Define the values: $S_0 = 0, \quad S_1, S_2, \dots, S_n \mod n$ Notice that there are $n+1$ sums in total ($S_0$ to $S_n$) and only $n$ possible remainders when divided by $n$ (which are $0, 1, 2, \dots, n-1$).

3. Pigeonhole Principle: By the pigeonhole principle, among the $n+1$ values $S_0, S_1, \dots, S_n$, at least two of them must have the same value modulo $n$. Suppose $S_j \equiv S_k \pmod{n}$ for some $j < k$.

4. Conclude the Solution: If $S_j \equiv S_k \pmod{n}$, then: $S_k - S_j \equiv 0 \pmod{n}$ which means that the sum of the elements from $a_{j+1}$ to $a_k$, i.e., $\sum_{i=j+1}^{k} a_i$ is divisible by $n$.

Therefore, we have shown that there exist integers $j$ and $k$ such that the sum of the sequence from $j$ to $k$ is a multiple of $n$, as required.

Would you like any further clarifications or detailed examples?

Additional Related Questions:

1. How does the pigeonhole principle work in problems involving modulo arithmetic?
2. Can you generalize this result to sequences where the number of terms exceeds $n$?
3. What are some other common applications of the pigeonhole principle in number theory?
4. Can this problem be solved using any other mathematical technique like induction?
5. What changes if the sequence $a_1, a_2, \dots, a_n$ contains negative numbers?

Tip: When working with modular arithmetic problems, always try to reduce sums or sequences modulo the divisor to simplify the calculations.