Folklore

$a_{1}, a_{2}, \ldots, a_{101}$ - is a permutation of the numbers $2,3, \ldots, 102$, such that $a_{k}$ is divisible by $k$ for each $k$. Find all such permutations.
Let's add $a_{102}=1$. We have obtained a permutation on the set $\{1,2, \ldots, 102\}$. It decomposes into cycles. But the smallest number in a non-trivial (containing more than one number) cycle stands at a position with a number greater than the number itself. According to the condition, such a number can only be 1, therefore, there is only one non-trivial cycle (and it consists of some divisors of the number 102, each of which is a divisor of the next), while the other cycles are trivial. Here are all the variants of non-trivial cycles (there are 13 of them):

(1, 102), (1, 2, 102), (1, 3, 102), (1, 6, 102), (1, 17, 102), (1, 34, 102), (1, 51, 102), (1, 2, 6, 102), (1,2,34,102), (1,3,6,102), (1,3,51,102), (1,17,34,102), (1,17,51,102).