`1!+2!+3!+4!+ ----100!` divided by 5 . find Remainder .

Answer

To find the remainder when \(1! + 2! + 3! + 4! + \ldots + 100!\) is divided by 5, we will first calculate the factorials from \(1!\) to \(4!\) and then analyze the behavior of factorials from \(5!\) onward.

### Step-by-step Solution:

1. **Calculate the factorials:**
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)

2. **Sum the factorials from \(1!\) to \(4!\):**
\[ 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33 \]

3. **Calculate the factorials from \(5!\) onward:**
- \(5! = 120\)
- \(6! = 720\)
- \(7! = 5040\)
- and so on...

Notice that all factorials from \(5!\) onward are multiples of \(5\). This means they will contribute \(0\) to the remainder when divided by \(5\).

4. **Combine the results:**
Since \(5!, 6!, 7!, \ldots, 100!\) all give a remainder of \(0\) when divided by \(5\), we only need to consider the sum \(1! + 2! + 3! + 4!\).

5. **Find the remainder of \(33\) when divided by \(5\):**
\[ 33 \div 5 = 6 \quad \text{(remainder is 3)} \]

Thus, the remainder when \(1! + 2! + 3! + 4! + \ldots + 100!\) is divided by \(5\) is \(3\).

### Final Answer:
The remainder is \(3\).