When `2^21` is divided by 9, the remainder obtained is ?

Options

A. a) 1
B. b) 2
C. c) 8
D. d) 6

Answer

To find the remainder when \(2^{21}\) is divided by 9, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Express \(2^{21}\) in a manageable form
We can express \(2^{21}\) as \((2^3)^7 \cdot 2^0\) since \(21 = 3 \times 7\). This helps us simplify our calculations.

### Step 2: Calculate \(2^3\)
Calculate \(2^3\):
\[ 2^3 = 8 \]

### Step 3: Find \(8 \mod 9\)
Now, we need to find \(8\) modulo \(9\):
\[ 8 \mod 9 = 8 \]
Since \(8\) is less than \(9\), the remainder when \(8\) is divided by \(9\) is \(8\).

### Step 4: Raise to the power of 7
Now we need to find \((2^3)^7 \mod 9\):
\[ (2^3)^7 = 8^7 \]

### Step 5: Calculate \(8^7 \mod 9\)
Since \(8 \equiv -1 \mod 9\), we can rewrite:
\[ 8^7 \equiv (-1)^7 \mod 9 \]
Calculating this gives:
\[ (-1)^7 = -1 \]

### Step 6: Convert \(-1\) to a positive remainder
Now, convert \(-1\) to a positive remainder:
\[ -1 \mod 9 = 8 \]

### Conclusion
Thus, the remainder when \(2^{21}\) is divided by \(9\) is:
\[ \text{Remainder} = 8 \]