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# Factorial Sum!!!

What is the remainder when (1!+2!+3!+...+200!) is divided by 14.?

Note by Sanjay Kumar
4 years, 3 months ago

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For each $$n \geq 7$$, $$n!$$ is divisible by $$14$$ (as it is divisible by both $$2$$ and $$7$$), so we only have to consider the first six terms: $1! + 2! + 3! + 4! + 5! + 6! = 1 + 2 + 6 + 24 + 120 + 720 = 873 = 62 \cdot 14 + 5,$ so the answer is $$\boxed{5}$$.

- 4 years, 3 months ago