When `(9^8+1)` is divided by 8, the remainder will be?

Options

A. a)3
B. b)2
C. c)1
D. d)4

Answer

To find the remainder when \(9^8 + 1\) is divided by 8, we can follow these steps:

### Step 1: Simplify \(9\) modulo \(8\)
First, we need to find \(9\) modulo \(8\):
\[ 9 \div 8 = 1 \quad \text{(quotient)} \]
\[ 9 - (8 \times 1) = 1 \quad \text{(remainder)} \]
Thus, \(9 \equiv 1 \mod 8\).

### Step 2: Substitute \(9\) in the expression
Now we can substitute \(9\) with \(1\) in the expression \(9^8 + 1\):
\[ 9^8 + 1 \equiv 1^8 + 1 \mod 8 \]

### Step 3: Calculate \(1^8\)
Next, we calculate \(1^8\):
\[ 1^8 = 1 \]
Therefore,
\[ 1^8 + 1 \equiv 1 + 1 \mod 8 \]

### Step 4: Add the results
Now we add the results:
\[ 1 + 1 = 2 \]

### Step 5: Find the final remainder
Thus, when \(9^8 + 1\) is divided by \(8\), the remainder is:
\[ 2 \]

### Final Answer
The remainder when \(9^8 + 1\) is divided by \(8\) is \(2\).

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