31. For each number from 1 to 1000, all its natural divisors were listed (as a result, some numbers were listed multiple times). Determine which is greater: the sum of all listed numbers or one million?
31. Answer: a million is greater than the sum of the written numbers.

Let's add separately all the written ones, all the written twos, all the written threes, etc.

Consider any number $n$, where $1 \leqslant n \leqslant 1000$. It is a divisor of the numbers $n, 2 n, 3 n, \ldots, k n$, where $k n$ is the largest number not exceeding 1000 and divisible by $n$. Thus, $k=[1000 / n]$ and the number $n$ is written $[1000 / n]$ times. Therefore, the sum of all written instances of the number $n$ is $n \cdot[1000 / n] \leqslant 1000$.

In total, we have calculated 1000 sums, and each does not exceed 1000. However, in reality, many of the sums are less than 1000, for example, the number $n=3$ is written 333 times and contributes 999 to the total sum. Therefore, the final sum of all the numbers is less than a million.