Find the remiander when `32^(32)` is divided 6.

Options

A. 0
B. 5
C. 4
D. 2

Answer

To find the remainder when \( 32^{32} \) is divided by 6, we can follow these steps:

### Step 1: Simplify the base
First, we can simplify \( 32 \) modulo \( 6 \):
\[ 32 \div 6 = 5 \quad \text{(with a remainder of 2)} \]
Thus,
\[ 32 \equiv 2 \mod 6 \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression using the simplified base:
\[ 32^{32} \equiv 2^{32} \mod 6 \]

### Step 3: Find the pattern in powers of 2 modulo 6
Next, we need to find the powers of \( 2 \) modulo \( 6 \):
- \( 2^1 \equiv 2 \mod 6 \)
- \( 2^2 \equiv 4 \mod 6 \)
- \( 2^3 \equiv 8 \equiv 2 \mod 6 \)
- \( 2^4 \equiv 16 \equiv 4 \mod 6 \)

From this, we can see that the powers of \( 2 \) modulo \( 6 \) repeat every two terms:
- When the exponent is odd, \( 2^n \equiv 2 \mod 6 \)
- When the exponent is even, \( 2^n \equiv 4 \mod 6 \)

### Step 4: Determine whether \( 32 \) is odd or even
Since \( 32 \) is an even number, we can conclude:
\[ 2^{32} \equiv 4 \mod 6 \]

### Step 5: Conclusion
Thus, the remainder when \( 32^{32} \) is divided by \( 6 \) is:
\[ \text{Remainder} = 4 \]