Find the remainder when the number `9^(100)` is divided by 8.

Answer

To find the remainder when \( 9^{100} \) is divided by 8, we can use modular arithmetic. Here’s the step-by-step solution:

### Step 1: Simplify the base modulo 8
First, we simplify \( 9 \) modulo \( 8 \):
\[ 9 \mod 8 = 1 \]
This means that \( 9 \) is equivalent to \( 1 \) when divided by \( 8 \).

### Step 2: Substitute the simplified base into the exponent
Now we can replace \( 9 \) with \( 1 \) in our original expression:
\[ 9^{100} \mod 8 \equiv 1^{100} \mod 8 \]

### Step 3: Evaluate the exponent
Since any number raised to any power is still that number (as long as the base is not zero), we have:
\[ 1^{100} = 1 \]

### Step 4: Find the remainder
Now we find the remainder when \( 1 \) is divided by \( 8 \):
\[ 1 \mod 8 = 1 \]

### Conclusion
Thus, the remainder when \( 9^{100} \) is divided by \( 8 \) is:
\[ \boxed{1} \]