Find the remainder when 1! + 2! + 3! + 4! + 5! + .......100! is divided by 24

Key insight: For n ≥ 4, n! is divisible by 24 because 4! = 24 and any larger factorial contains 4! as a factor.

• Therefore all terms from 4! up to 100! are multiples of 24 and contribute 0 to the remainder.

• Only 1!, 2!, and 3! can contribute nonzero remainders when dividing by 24.

Compute the small factorials:

• 1! = 1

• 2! = 2

• 3! = 6

Sum and remainder:

1 + 2 + 6 = 9, so the sum 1! + 2! + 3! + ... + 100! leaves a remainder of 9 when divided by 24.