We take 4 consecutive integers such that
((1 !) + (2 !) + (3 !) + (4 !)) / (1 * 2 * 3 * 4) = 1.37500
((2 !) + (3 !) + (4 !) + (5 !)) / (2 * 3 * 4 * 5) = 1.26666667
((3 !) + (4 !) + (5 !) + (6 !)) / (3 * 4 * 5 * 6) = 2.41666667
((4 !) + (5 !) + (6 !) + (7 !)) / (4 * 5 * 6 * 7) = 7.02857143
((5 !) + (6 !) + (7 !) + (8 !)) / (5 * 6 * 7 * 8) = 27.5
((6 !) + (7 !) + (8 !) + (9 !)) / (6 * 7 * 8 * 9) = 135.238095
((7 !) + (8 !) + (9 !) + (10 !)) / (7 * 8 * 9 * 10) = 801
We get our first integer
If we continue onto the next series, we get
((8 !) + (9 !) + (10 !) + (11 !)) / (8 * 9 * 10 * 11) = 5549.09091
This is not an integer. Could you predict when we could get an integer?
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