Problem 1076

For a positive integer $n$, let $\operatorname{rad}$ n be the product of the distinct prime factors of $n$. Observe that if $n \mid m^{m}$, all prime factors of $n$ must divide $m$, so $\operatorname{rad} n \mid$ m. Therefore, if $n$ is such an integer, $\operatorname{rad}$ n must divide at least two of the numbers in $\{1,2,3,4,5,6,7,8,9,10\}$, implying that rad $n$ is either $1,2,3$, or 5. These have $1,10,6$, and 5 cases, respectively, for a total of 22.