Given any set of $15$ integers, must there be two that have the same remainder when divided by $12?$ Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Either describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co$-$domain, and how elements of the domain are related to elements of the co$-$domain.