How many factorials of non-negative integers are in the set ?
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Since the answer is 1 greater than a multiple of 7, it cannot be divisible by 7. Therefore, we only need to check all of them up to 6.
0 ! ≡ 1 m o d 7
1 ! ≡ 1 m o d 7
2 ! ≡ 2 ∗ 1 m o d 7 ≡ 2 m o d 7
3 ! ≡ 3 ∗ 2 m o d 7 ≡ 6 m o d 7
4 ! ≡ 4 ∗ 6 m o d 7 ≡ 3 m o d 7
5 ! ≡ 5 ∗ 3 m o d 7 ≡ 1 m o d 7
6 ! ≡ 6 ∗ 1 m o d 7 ≡ 6 m o d 7
Therefore, the only ones that work are 0, 1, and 5. So the answer is 3 .