Find the remainder of `(17^(18^(19^(20^(….)))))/8`

Options

A. 2
B. 3
C. 7
D. 1

Answer

To find the remainder of \( \frac{17^{18^{19^{20^{\ldots}}}}}{8} \), we can follow these steps:

### Step 1: Find the remainder of 17 when divided by 8
First, we need to determine \( 17 \mod 8 \).

\[ 17 \div 8 = 2 \quad \text{(since } 8 \times 2 = 16\text{)} \]
\[ \text{Remainder} = 17 - 16 = 1 \]

### Step 2: Rewrite the expression using the remainder
Since \( 17 \equiv 1 \mod 8 \), we can rewrite the expression:

\[ 17^{18^{19^{20^{\ldots}}}} \equiv 1^{18^{19^{20^{\ldots}}}} \mod 8 \]

### Step 3: Evaluate \( 1^{18^{19^{20^{\ldots}}}} \)
Now, we know that any number raised to any power will still be 1:

\[ 1^{18^{19^{20^{\ldots}}}} = 1 \]

### Step 4: Find the final remainder
Thus, we have:

\[ 17^{18^{19^{20^{\ldots}}}} \equiv 1 \mod 8 \]

### Conclusion
The remainder when \( 17^{18^{19^{20^{\ldots}}}} \) is divided by 8 is:

\[ \boxed{1} \]