Question

Find the unit digit of \(1! + 2! + 3! + \cdots + 100!\).

Hint:

Factorials from \(5!\) onwards always have unit digit zero, so they do not affect unit digit calculations.

Answer 1

Step 1: Observe the pattern of unit digits of factorials.
\[ 1! = 1,\quad 2! = 2,\quad 3! = 6,\quad 4! = 24 \]
From \(5!\) onwards, all factorials end with 0 in the unit digit.
Step 2: Consider only relevant terms.
So, the unit digits contributing to the sum are:
\[ 1! + 2! + 3! + 4! = 1 + 2 + 6 + 4 = 13 \]
Step 3: Find the unit digit.
The unit digit of 13 is 3.
Step 4: Conclusion.
The unit digit of \(1! + 2! + 3! + \cdots + 100!\) is 3.