What is the least positive integer divisible by each of the first five composite numbers?
The first five composite numbers are 4, 6, 8, 9, and 10. Their prime factorizations are $2^2, 2\cdot 3, 2^3, 3^2$, and $2\cdot 5$. Taking the maximum exponent for each prime, we find that the least common multiple is $2^3\cdot 3^2\cdot 5=\boxed{360}$.