The remainder when `8^(1785)` is divided by 7 is :

Options

A. 5
B. 1
C. 6
D. can't be determined

Answer

To find the remainder when \( 8^{1785} \) is divided by 7, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Rewrite the base in terms of modulo 7
First, we can simplify the base \( 8 \) modulo \( 7 \):
\[ 8 \mod 7 = 1 \]
So, we can rewrite \( 8^{1785} \) as:
\[ 8^{1785} \equiv 1^{1785} \mod 7 \]

### Step 2: Calculate \( 1^{1785} \)
Next, we calculate \( 1^{1785} \):
\[ 1^{1785} = 1 \]

### Step 3: Find the remainder
Now we substitute back into our equation:
\[ 8^{1785} \equiv 1 \mod 7 \]
This means that when \( 8^{1785} \) is divided by \( 7 \), the remainder is \( 1 \).

### Conclusion
Thus, the remainder when \( 8^{1785} \) is divided by \( 7 \) is:
\[ \text{Remainder} = 1 \]

---