1.

Write the remainder obtained when 1! + 2! + 3! + ..... + 200! is divided by 14

Answer» Here, the given expression is,
`1!+2!+3!+...+200!`
`1! = 1`
`2! = 2**1=2`
`3! = 3**2**1 = 6`
`4! = 4**3**2**1 = 24`
`5! = 5**4**3**2**1 = 120`
`6! = 6**5**4**3**2**1 = 720`
`7! = 7**6**5**4**3**2**1 = 5040`
So, `7!` is divisible by `14`.
Now, `8! = 8**7!`
So, `8!` will also be divisible by `14`.Similarly, every term greater than `7!` in the given expression will be divisible by `14`.
So, sum of the terms that are not divisible by `14` is,
`1!+2!+3!+4!+5!+6! = 1+2+6+24+120+720 = 873`
`:.` Remainder of `873` when divided by `14` will be `5` which is the required answer.


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