Find the remainder when `11^8` is divided by 7.

Answer

To find the remainder when \( 11^8 \) is divided by 7, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Simplify \( 11 \mod 7 \)
First, we need to find \( 11 \mod 7 \):
\[ 11 \div 7 = 1 \quad \text{(quotient)} \]
\[ 11 - (7 \times 1) = 11 - 7 = 4 \]
So, \( 11 \equiv 4 \mod 7 \).

### Step 2: Rewrite the expression
Now we can rewrite \( 11^8 \) in terms of modulo 7:
\[ 11^8 \equiv 4^8 \mod 7 \]

### Step 3: Calculate \( 4^8 \mod 7 \)
Next, we need to calculate \( 4^8 \mod 7 \). We can use the property of exponents:
\[ 4^2 = 16 \]
Now, find \( 16 \mod 7 \):
\[ 16 \div 7 = 2 \quad \text{(quotient)} \]
\[ 16 - (7 \times 2) = 16 - 14 = 2 \]
So, \( 4^2 \equiv 2 \mod 7 \).

### Step 4: Continue calculating powers
Now we find \( 4^4 \):
\[ 4^4 = (4^2)^2 \equiv 2^2 \mod 7 \]
Calculating \( 2^2 \):
\[ 2^2 = 4 \]
So, \( 4^4 \equiv 4 \mod 7 \).

### Step 5: Calculate \( 4^8 \)
Now we find \( 4^8 \):
\[ 4^8 = (4^4)^2 \equiv 4^2 \mod 7 \]
We already calculated \( 4^2 \equiv 2 \mod 7 \).

### Final Step: Conclusion
Thus, \( 11^8 \equiv 4^8 \equiv 2 \mod 7 \).

Therefore, the remainder when \( 11^8 \) is divided by 7 is **2**.

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