The remainder when the number `3^(256) - 3^(12)` is divided by 8 is (a) 0 (b) 3 (c) 4 (d) 7

Options

A. 0
B. 3
C. 4
D. 7

Explanation

To find the remainder when \(3^{256} - 3^{12}\) is divided by 8, we can follow these steps:

### Step 1: Factor out \(3^{12}\)
We can rewrite the expression:
\[ 3^{256} - 3^{12} = 3^{12}(3^{244} - 1) \]

### Step 2: Analyze \(3^{12} \mod 8\)
Next, we need to find \(3^{12} \mod 8\). We can use the property of powers of 3 modulo 8.

Calculating the first few powers of 3 modulo 8:
- \(3^1 \equiv 3 \mod 8\)
- \(3^2 \equiv 9 \equiv 1 \mod 8\)

Since \(3^2 \equiv 1 \mod 8\), we can see that every even power of 3 will also be congruent to 1 modulo 8. Therefore:
\[ 3^{12} = (3^2)^6 \equiv 1^6 \equiv 1 \mod 8 \]

### Step 3: Analyze \(3^{244} - 1 \mod 8\)
Now we need to find \(3^{244} \mod 8\). Since \(244\) is even, we can use the same reasoning:
\[ 3^{244} = (3^2)^{122} \equiv 1^{122} \equiv 1 \mod 8 \]
Thus:
\[ 3^{244} - 1 \equiv 1 - 1 \equiv 0 \mod 8 \]

### Step 4: Combine results
Now substituting back into our expression:
\[ 3^{256} - 3^{12} = 3^{12}(3^{244} - 1) \equiv 1 \cdot 0 \equiv 0 \mod 8 \]

### Conclusion
The remainder when \(3^{256} - 3^{12}\) is divided by 8 is:
\[ \boxed{0} \]