What is the Remainder?

What is the remainder left after dividing 1! + 2! + 3! + … + 100! By7?

Answer:

From 7! onwards all the terms are divisible by 7 as 7 is one of the factor. So there is no remainder left for those terms. i.e. remainder left after dividing 7! + 8! + 9! + ... + 100! is 0. The only part to be consider is 1! + 2! + 3! + 4! + 5! + 6! = 1 + 2 + 6 + 24 + 120 + 720= 873. The remainder left after dividing 873 by 7 is 5. Hence the remainder is 5.