jmc

Since $1$ divides every integer, we need only find the least common multiple (LCM) of the numbers $2$ through $10$. Their prime factorizations are, respectively, $2,3,2^2,5,2\cdot 3,7,2^3,3^2,2\cdot 5$. In its own prime factorization, the LCM must have each prime that appears in this list, raised to at minimum the highest power that appears in the list for that prime, in order for it to divide all of the integers from $2$ through $10$. Thus, the prime factorization of the LCM is $2^3\cdot 3^2\cdot 5\cdot 7$. So the LCM is $8\cdot 9\cdot 5\cdot 7=\boxed{2520}$.