jmc

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 $2^3\cdot 3^2\cdot 5=\boxed{360}$.