If I'm not mistaken, N! | NN, as N! = 1*2*...*N, and NN = N*N*...*N. Therefore, NN/N! = NN-1/(N-1)!
The fact that N N and N! share a factor doesn't mean that N! is a divisor of N N, as the following table shows:
N NN N! NN % N!
1 1 1 0
2 4 2 0
3 27 6 3
4 256 24 16
5 3125 120 5
6 46656 720 576
7 823543 5040 2023
8 16777216 40320 4096
9 387420489 362880 227529
10 10000000000 3628800 2656000
11 285311670611 39916800 26301011
12 8916100448256 479001600 443667456
13 302875106592253 6227020800 5268921853
14 11112006825558016 87178291200 294332416
15 437893890380859375 1307674368000 820814907375
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