The pigeonhole principle, a seemingly trivial observation, is a cornerstone of combinatorial number theory. It directly implies that among any 10 random integers, two must have the same remainder when divided by 9—a fact with far-reaching consequences, such as proving that any set of 5 integers contains 3 that sum to a multiple of 3, and it underlies the Erdős–Ginzburg–Ziv theorem that any 2n−1 integers contain n whose sum is divisible by n.