Question

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

Answer 1

First we will find the least factorial term divisible by 14

As 7! = $7\times 6\times 5!$ is divisible by 14 leaving remainder zero.

Hence terms 7! onwards can be written as multiple of 7!

8! =$8\times 7!,9!=9\times 8\times 7!$ ... like ways 200! can also be wirtten as multiple of 7!

So all the terms 7! onwards are divisible by 14 leaving remainder zero.

1!+2!+3!+4!+5!+6!

= 1+2+6+24+120+720

= 873

Hence, remainder obtained when 1! + 2! +3!+...+200! is divided by 14 is 5.