Find the remainder when `4^(5^(6^(7^(8^(9^(10))))))` is divided by 6.

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the remainder when \( 4^{5^{6^{7^{8^{9^{10}}}}}} \) is divided by 6, we can simplify the problem using properties of modular arithmetic.

### Step-by-Step Solution:

1. **Understanding the Base Modulo 6**:
We start by observing the base \( 4 \) modulo \( 6 \):
\[ 4 \mod 6 = 4 \]
This means that \( 4 \) is less than \( 6 \) and gives a remainder of \( 4 \).

2. **Finding Powers of 4 Modulo 6**:
Next, we will compute the first few powers of \( 4 \) modulo \( 6 \):
- \( 4^1 \mod 6 = 4 \)
- \( 4^2 = 16 \) and \( 16 \mod 6 = 4 \)
- \( 4^3 = 64 \) and \( 64 \mod 6 = 4 \)

From these calculations, we can see that:
\[ 4^n \mod 6 = 4 \quad \text{for any positive integer } n \]

3. **Applying the Result to the Given Expression**:
The expression we need to evaluate is \( 4^{5^{6^{7^{8^{9^{10}}}}}} \). Since we have established that \( 4^n \mod 6 = 4 \) for any positive integer \( n \), we can conclude:
\[ 4^{5^{6^{7^{8^{9^{10}}}}}} \mod 6 = 4 \]

4. **Final Result**:
Therefore, the remainder when \( 4^{5^{6^{7^{8^{9^{10}}}}}} \) is divided by \( 6 \) is:
\[ \boxed{4} \]