The factorials up to 1015 :
1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 6000 values, from 1 to 6000!).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 5999 | 1 | 2 | ||||||||
| 3 | 5998 | 1 | 1 | 3 | |||||||
| 4 | 5997 | 1 | 2 | 0 | 4 | ||||||
| 5 | 5996 | 2 | 1 | 0 | 1 | 5 | |||||
| 6 | 5998 | 1 | 1 | 0 | 0 | 0 | 6 | ||||
| 7 | 5994 | 2 | 1 | 1 | 0 | 0 | 2 | 7 | |||
| 8 | 5997 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 8 | ||
| 9 | 5995 | 1 | 1 | 1 | 0 | 0 | 2 | 0 | 0 | 9 | |
| 10 | 5996 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 10 |
| 11 | 5990 | 2 | 3 | 0 | 0 | 2 | 1 | 0 | 0 | 0 | 2 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.
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