Find the remainder in expression -
`(1!^(2) + 2!^(2) + ……. 100!^(2))/(5)`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the remainder of the expression \((1!^2 + 2!^2 + \ldots + 100!^2) \div 5\), we can simplify the problem by focusing on the first few factorials, as factorials greater than or equal to \(5!\) will be divisible by \(5\).

### Step-by-Step Solution:

1. **Identify Factorials**: We know that \(n!\) for \(n \geq 5\) will be divisible by \(5\). Thus, we only need to calculate \(1!^2\), \(2!^2\), \(3!^2\), and \(4!^2\).

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

3. **Square the Factorials**:
- \(1!^2 = 1^2 = 1\)
- \(2!^2 = 2^2 = 4\)
- \(3!^2 = 6^2 = 36\)
- \(4!^2 = 24^2 = 576\)

4. **Sum the Squares**:
\[ 1!^2 + 2!^2 + 3!^2 + 4!^2 = 1 + 4 + 36 + 576 \]
\[ = 617 \]

5. **Find the Remainder when Divided by 5**:
To find \(617 \mod 5\), we can divide \(617\) by \(5\):
\[ 617 \div 5 = 123 \quad \text{(quotient)} \]
\[ 123 \times 5 = 615 \quad \text{(product)} \]
\[ 617 - 615 = 2 \quad \text{(remainder)} \]

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