For any positive integer n, if `4^(n) -3n` is divided by 9, then what is the remainder?
(a)8
(b)6
(c)1
(d)4

Options

A. 8
B. 6
C. 4
D. 1

Answer

To solve the problem of finding the remainder when \( 4^n - 3n \) is divided by 9 for any positive integer \( n \), we will evaluate the expression for different values of \( n \) and look for a pattern.

### Step-by-Step Solution:

1. **Evaluate for \( n = 1 \)**:
\[ 4^1 - 3 \cdot 1 = 4 - 3 = 1 \]
Now, divide 1 by 9:
\[ 1 \div 9 = 0 \quad \text{(remainder 1)} \]

2. **Evaluate for \( n = 2 \)**:
\[ 4^2 - 3 \cdot 2 = 16 - 6 = 10 \]
Now, divide 10 by 9:
\[ 10 \div 9 = 1 \quad \text{(remainder 1)} \]

3. **Evaluate for \( n = 3 \)**:
\[ 4^3 - 3 \cdot 3 = 64 - 9 = 55 \]
Now, divide 55 by 9:
\[ 55 \div 9 = 6 \quad \text{(remainder 1)} \]

4. **Evaluate for \( n = 4 \)**:
\[ 4^4 - 3 \cdot 4 = 256 - 12 = 244 \]
Now, divide 244 by 9:
\[ 244 \div 9 = 27 \quad \text{(remainder 1)} \]

5. **Conclusion**:
From the evaluations above, we can see that for \( n = 1, 2, 3, \) and \( 4 \), the remainder when \( 4^n - 3n \) is divided by 9 is consistently 1.

Thus, the remainder when \( 4^n - 3n \) is divided by 9 for any positive integer \( n \) is:
\[ \boxed{1} \]