Remainder of the sum of factorials when divided by 24

We are asked to find the remainder when \(X = 1! + 2! + 3! + \cdots + 100!\) is divided by 24.

For factorials where \(n \geq 4\), \(n!\) is divisible by 24, because \(4! = 24\). Hence, the sum simplifies to:

\(X = 1! + 2! + 3!\)

Now, calculating the factorials:

\(1! = 1\), \(2! = 2\), \(3! = 6\)

Thus, \(1! + 2! + 3! = 1 + 2 + 6 = 9\)

Now, we calculate the remainder when 9 is divided by 24:

\(9 \mod 24 = 9\)

Therefore, the remainder when \(X\) is divided by 24 is 9.