16. Given 2024 sets, each consisting of 44 elements. The union of any two of these sets contains 87 elements. How many elements does the union of all 2024 sets contain?
Answer: 87033.

II. Find the number of natural numbers $n$, not exceeding 500, for which the equation $x^{[x]}=n$ has a solution. Here $[x]$ is the greatest integer not exceeding $x$.

Solution. If $[x]=0$, then the solution is: $0 \leqslant x < 1$, and $n=1$.

If $[x]=1$, then the solution is: $1 \leqslant x < 2$, and $n=1$.

If $[x]=2$, then the solution is: $2 \leqslant x < 3$, and $n=4, 5, 6, 7, 8$.

If $[x]=3$, then the solution is: $3 \leqslant x < 4$, and $n=27, 28, \ldots, 63$.

If $[x]=4$, then the solution is: $4 \leqslant x < 5$, and $n=256, 257, \ldots, 500$.

Therefore, we get: $n=1,4,5,6,7,8,27,28, \ldots, 62,63,256,257, \ldots, 500$. In total, there are 288 numbers.

Answer: 288.