78*. Find all integers $n$ that are divisible by all integers not exceeding $\sqrt{n}$.
78. Suppose that the number $n$ is divisible by all numbers $m$ less than or equal to $\sqrt{n}$. Let's form the least common multiple $K$ of all such numbers $m$. Clearly, this will include all prime numbers less than $\sqrt{n}$, and each prime number $p$ will be raised to a power $k$ such that $p^k \leq \sqrt{n}$, but $p^{k+1} > \sqrt{n}$. Suppose the number of prime numbers less than $\sqrt{n}$ is $l$; we will denote these prime numbers as $p_1, p_2, \ldots, p_l$. The least common multiple $K$ of all numbers less than $\sqrt{n}$ is the product $p_1^{k_1} p_2^{k_2} \ldots p_l^{k_l}$, where $k_1$ is such that $p_1^{k_1} \leq \sqrt{n}$ (the fourth prime number is 7) and $n < 49$.

By checking all numbers less than 49, we can easily verify that the numbers possessing the required property are $24, 12, 8, 6, 4$, and $2$.