Five consecutive composite integers

Statement

Find five consecutive composite integers, choosing the smallest possible block.

Solution

We look for five consecutive integers with no prime among them. Since $23$ and $29$ are prime, there is just enough room between them: $\boxed{24,\;25,\;26,\;27,\;28}.$ Indeed $24=2^3\cdot 3$, $25=5^2$, $26=2\cdot 13$, $27=3^3$, $28=2^2\cdot 7$: all composite. It is the smallest block, because any run of five starting earlier meets a prime.