When `2^(301)` is divided by 5, the least positive remainder is

Options

A. 4
B. 8
C. 2
D. 6

Answer

To find the least positive remainder when \(2^{301}\) is divided by 5, we can use properties of modular arithmetic. Here's a step-by-step solution:

### Step 1: Use Modular Arithmetic
We need to find \(2^{301} \mod 5\).

### Step 2: Identify a Pattern
First, let's calculate the powers of 2 modulo 5:
- \(2^1 \mod 5 = 2\)
- \(2^2 \mod 5 = 4\)
- \(2^3 \mod 5 = 3\)
- \(2^4 \mod 5 = 1\)

Notice that \(2^4 \equiv 1 \mod 5\). This means that every fourth power of 2 will cycle back to 1.

### Step 3: Reduce the Exponent
Since \(2^4 \equiv 1 \mod 5\), we can reduce the exponent \(301\) modulo \(4\) (the cycle length):
\[ 301 \mod 4 \]
Calculating this, we find:
\[ 301 = 4 \times 75 + 1 \quad \Rightarrow \quad 301 \mod 4 = 1 \]

### Step 4: Calculate the Result
Now we can substitute back into our modular equation:
\[ 2^{301} \mod 5 \equiv 2^{1} \mod 5 \]
Thus, we have:
\[ 2^{301} \mod 5 \equiv 2 \]

### Conclusion
The least positive remainder when \(2^{301}\) is divided by 5 is \(2\).