When `2^34` is divided by 9 , the remainder will be?

Options

A. a) 7
B. b) 8
C. c) 5
D. d) 6

Answer

To find the remainder when \( 2^{34} \) is divided by 9, we can use the concept of modular arithmetic and properties of exponents. Here’s a step-by-step solution:

### Step 1: Identify the problem
We need to find \( 2^{34} \mod 9 \).

### Step 2: Use properties of exponents
Instead of calculating \( 2^{34} \) directly, we can look for a pattern in the powers of 2 modulo 9.

### Step 3: Calculate the first few powers of 2 modulo 9
- \( 2^1 = 2 \)
- \( 2^2 = 4 \)
- \( 2^3 = 8 \)
- \( 2^4 = 16 \equiv 7 \mod 9 \) (since \( 16 - 9 = 7 \))
- \( 2^5 = 32 \equiv 5 \mod 9 \) (since \( 32 - 27 = 5 \))
- \( 2^6 = 64 \equiv 1 \mod 9 \) (since \( 64 - 63 = 1 \))

### Step 4: Identify the cycle
From the calculations, we see that \( 2^6 \equiv 1 \mod 9 \). This means every sixth power of 2 will repeat the cycle:
- \( 2^1 \equiv 2 \)
- \( 2^2 \equiv 4 \)
- \( 2^3 \equiv 8 \)
- \( 2^4 \equiv 7 \)
- \( 2^5 \equiv 5 \)
- \( 2^6 \equiv 1 \)

### Step 5: Reduce the exponent modulo 6
Since \( 2^6 \equiv 1 \mod 9 \), we can reduce \( 34 \) modulo \( 6 \):
\[ 34 \div 6 = 5 \quad \text{(with a remainder of 4)} \]
Thus, \( 34 \mod 6 = 4 \).

### Step 6: Find \( 2^4 \mod 9 \)
From our earlier calculations:
\[ 2^4 \equiv 7 \mod 9 \]

### Conclusion
Therefore, the remainder when \( 2^{34} \) is divided by 9 is \( 7 \).

### Final Answer
The remainder is \( 7 \).

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