`(1!+2!+----+100!)/(7)` Find R.

Answer

To solve the problem \((1! + 2! + 3! + \ldots + 100!) / 7\) and find the remainder \(R\), we can follow these steps:

### Step 1: Understand Factorials
Factorials are defined as follows:
- \(1! = 1\)
- \(2! = 1 \times 2 = 2\)
- \(3! = 1 \times 2 \times 3 = 6\)
- \(4! = 1 \times 2 \times 3 \times 4 = 24\)
- \(5! = 1 \times 2 \times 3 \times 4 \times 5 = 120\)
- \(6! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 = 720\)
- \(7! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 = 5040\)

### Step 2: Identify Factorials Divisible by 7
Notice that starting from \(7!\), all factorials will be divisible by \(7\) because they include \(7\) as a factor. Therefore, we only need to consider \(1!\) through \(6!\) for our calculations.

### Step 3: Calculate the Sum of Factorials from 1 to 6
Now, we calculate the sum of the factorials from \(1!\) to \(6!\):
\[ 1! + 2! + 3! + 4! + 5! + 6! = 1 + 2 + 6 + 24 + 120 + 720 \]

Calculating this step-by-step:
- \(1 + 2 = 3\)
- \(3 + 6 = 9\)
- \(9 + 24 = 33\)
- \(33 + 120 = 153\)
- \(153 + 720 = 873\)

So, the sum \(S = 873\).

### Step 4: Divide by 7 and Find the Remainder
Next, we need to find the remainder when \(873\) is divided by \(7\):
\[ 873 \div 7 = 124 \quad \text{(integer part)} \]
Calculating \(124 \times 7\):
\[ 124 \times 7 = 868 \]
Now, subtract this from \(873\) to find the remainder:
\[ 873 - 868 = 5 \]

### Conclusion
Thus, the remainder \(R\) when \((1! + 2! + 3! + \ldots + 100!)\) is divided by \(7\) is:
\[ \boxed{5} \]