Cats sharing 999919 mice equally

Several cats (at least two) between them killed $999919$ mice, every cat accounting for the same number of mice, and that per-cat tally was larger than one. In addition, the per-cat tally exceeded the number of cats. Determine the number of cats.

991. The number of cats times mice per cat is $999919=1000^2-81^2=(1000-9)(1000+9)=991\times1009$, and both factors are prime. With both counts above one and mice per cat larger, there were $991$ cats each killing $1009$ mice.