Erdős–Ginzburg–Ziv theorem

Remarks

The proof first treats a prime $n = p$, using the Chevalley–Warning theorem or the Cauchy–Davenport theorem. For composite $n = ab$, the cases $a$ and $b$ imply the case $n$. One repeatedly extracts $2b - 1$ disjoint groups of $a$ elements, each with sum divisible by $a$. Then one applies the case $b$ to the $2b - 1$ group sums divided by $a$. The number $2n - 1$ cannot be lowered. The $2n - 2$ integers consisting of $n - 1$ zeros and $n - 1$ ones have no $n$ elements with sum divisible by $n$. Indeed, any $n$ of them contain $j$ ones with $1 \le j \le n - 1$, so their sum $j$ is not a multiple of $n$. For $n = 2$: among any three integers, two have the same parity, so their sum is even.

Going deeper

For a student

The theorem is the starting point of zero-sum theory in additive combinatorics. In the language of the integers modulo $n$, every sequence of $2n - 1$ elements of $\Z/n\Z$ has a subsequence of length $n$ with sum $0$. Related invariants, such as the Davenport constant, ask for the shortest length that forces a nonempty zero-sum subsequence.