The remainder `lfloor1+lfloor2+lfloor3+"……"+lfloor200` is divided by 14 is

Options

A. 3
B. 4
C. 5
D. None of these

Answer

To find the remainder when the sum \(1! + 2! + 3! + \ldots + 200!\) is divided by 14, we can follow these steps:

### Step 1: Understand Factorials and Their Divisibility
Factorials grow very quickly, and for \(n \geq 7\), \(n!\) will be divisible by 14 because \(14 = 2 \times 7\). Therefore, we only need to consider the factorials from \(1!\) to \(6!\).

### Step 2: Calculate Factorials from 1 to 6
Now, let's calculate the factorials:
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)
- \(6! = 720\)

### Step 3: Sum the Factorials
Now, we will sum these factorials:
\[ 1! + 2! + 3! + 4! + 5! + 6! = 1 + 2 + 6 + 24 + 120 + 720 \]

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

So, the total sum is \(873\).

### Step 4: Find the Remainder When Divided by 14
Next, we need to find the remainder of \(873\) when divided by \(14\):
\[ 873 \div 14 = 62 \quad \text{(whole number part)} \]
Calculating the product:
\[ 62 \times 14 = 868 \]
Now, subtract this from \(873\) to find the remainder:
\[ 873 - 868 = 5 \]

### Conclusion
Thus, the remainder when \(1! + 2! + 3! + \ldots + 200!\) is divided by \(14\) is \(5\).