Statement 1: Remainder when `3456^(2222)` is divided by 7 is 4.
Statement 2: Remainder when `5^(2222)` is divided by 7 is 4.

Options

A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of
B. Both the statements are TRUE but STATEMENT 2 is NOT the correct explanation of
C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE.
D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer

To solve the problem, we need to evaluate both statements regarding the remainders when certain powers are divided by 7.

### Step 1: Evaluate Statement 1
**Statement 1:** Remainder when \( 3456^{2222} \) is divided by 7.

1. **Reduce \( 3456 \) modulo \( 7 \)**:
\[ 3456 \div 7 = 493 \quad \text{(integer part)} \]
\[ 3456 - (493 \times 7) = 3456 - 3451 = 5 \]
Therefore, \( 3456 \equiv 5 \mod 7 \).

2. **Rewrite the expression**:
\[ 3456^{2222} \equiv 5^{2222} \mod 7 \]

3. **Use Fermat's Little Theorem**:
Fermat's Little Theorem states that if \( p \) is a prime and \( a \) is not divisible by \( p \), then:
\[ a^{p-1} \equiv 1 \mod p \]
Here, \( p = 7 \) and \( a = 5 \). Thus:
\[ 5^{6} \equiv 1 \mod 7 \]

4. **Reduce the exponent \( 2222 \) modulo \( 6 \)** (since \( 6 = 7 - 1 \)):
\[ 2222 \div 6 = 370 \quad \text{(integer part)} \]
\[ 2222 - (370 \times 6) = 2222 - 2220 = 2 \]
Therefore, \( 2222 \equiv 2 \mod 6 \).

5. **Calculate \( 5^{2222} \mod 7 \)**:
\[ 5^{2222} \equiv 5^{2} \mod 7 \]
\[ 5^{2} = 25 \]
\[ 25 \div 7 = 3 \quad \text{(integer part)} \]
\[ 25 - (3 \times 7) = 25 - 21 = 4 \]
Thus, the remainder when \( 3456^{2222} \) is divided by \( 7 \) is \( 4 \).

### Step 2: Evaluate Statement 2
**Statement 2:** Remainder when \( 5^{2222} \) is divided by 7.

1. **Use the result from Statement 1**:
We already found that:
\[ 5^{2222} \equiv 5^{2} \mod 7 \]
And we calculated \( 5^{2} \equiv 4 \mod 7 \).

Thus, the remainder when \( 5^{2222} \) is divided by \( 7 \) is also \( 4 \).

### Conclusion
Both statements are correct:
- Remainder when \( 3456^{2222} \) is divided by \( 7 \) is \( 4 \).
- Remainder when \( 5^{2222} \) is divided by \( 7 \) is \( 4 \).

### Final Answer
Both statements are correct.

---