Find the remainder when the number represented by 22334 raised to the power `(1^(2)+2^(2)+....+66^(2))` is divided by 5 ?

Options

A. 1
B. 4
C. 2
D. None of these

Answer

To find the remainder when the number represented by \( 22334 \) raised to the power \( (1^2 + 2^2 + \ldots + 66^2) \) is divided by \( 5 \), we can follow these steps:

### Step 1: Calculate the sum of squares
We need to find the sum of squares from \( 1^2 \) to \( 66^2 \). We can use the formula for the sum of squares:

\[ \text{Sum} = \frac{n(n + 1)(2n + 1)}{6} \]

where \( n = 66 \).

### Step 2: Substitute \( n = 66 \) into the formula
Substituting \( n = 66 \):

\[ \text{Sum} = \frac{66(66 + 1)(2 \times 66 + 1)}{6} \]

Calculating each part:

- \( 66 + 1 = 67 \)
- \( 2 \times 66 + 1 = 133 \)

Now substituting these values back into the formula:

\[ \text{Sum} = \frac{66 \times 67 \times 133}{6} \]

### Step 3: Calculate the product
Now we calculate \( 66 \times 67 \times 133 \):

1. First, calculate \( 66 \times 67 = 4422 \).
2. Next, calculate \( 4422 \times 133 = 587706 \).

### Step 4: Divide by 6
Now we divide \( 587706 \) by \( 6 \):

\[ \text{Sum} = \frac{587706}{6} = 97951 \]

### Step 5: Find the unit digit of \( 22334^{97951} \)
Next, we need to find the unit digit of \( 22334^{97951} \). The unit digit of \( 22334 \) is \( 4 \).

### Step 6: Determine the pattern of unit digits for powers of \( 4 \)
The unit digits of powers of \( 4 \) follow a pattern:
- \( 4^1 = 4 \) (unit digit is \( 4 \))
- \( 4^2 = 16 \) (unit digit is \( 6 \))
- \( 4^3 = 64 \) (unit digit is \( 4 \))
- \( 4^4 = 256 \) (unit digit is \( 6 \))

The pattern alternates between \( 4 \) and \( 6 \):
- Odd powers of \( 4 \) have a unit digit of \( 4 \).
- Even powers of \( 4 \) have a unit digit of \( 6 \).

### Step 7: Determine if \( 97951 \) is odd or even
Since \( 97951 \) is odd, the unit digit of \( 4^{97951} \) is \( 4 \).

### Step 8: Find the remainder when divided by \( 5 \)
Finally, we find the remainder of \( 4 \) when divided by \( 5 \):

\[ 4 \mod 5 = 4 \]

### Conclusion
The remainder when \( 22334^{(1^2 + 2^2 + \ldots + 66^2)} \) is divided by \( 5 \) is \( 4 \).

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