Problem - 309
Prove: randomly select 51 numbers from $1, 2, 3, \cdots, 100$, at least two of them must be relatively prime to each other.
Consider the following $50$ buckets: $$\{1, 2\}, \{3,4\}, ...., \{99, 100\}$$
By the pigeonhole principle, at least two of these $51$ numbers must be in the same bucket. It is well-known that two consecutive integers must be co-prime to each other.