Find the remainder when `(2)^(51)` is divided by 5.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the remainder when \(2^{51}\) is divided by 5, we can use the concept of modular arithmetic. Here’s a step-by-step solution:

### Step 1: Identify the pattern of powers of 2 modulo 5
We start by calculating the first few powers of 2 and their remainders when divided by 5:

- \(2^1 = 2\) → Remainder is 2
- \(2^2 = 4\) → Remainder is 4
- \(2^3 = 8\) → Remainder is 3 (since \(8 \mod 5 = 3\))
- \(2^4 = 16\) → Remainder is 1 (since \(16 \mod 5 = 1\))

Now, we notice that the remainders repeat every 4 powers:
- \(2^1 \mod 5 = 2\)
- \(2^2 \mod 5 = 4\)
- \(2^3 \mod 5 = 3\)
- \(2^4 \mod 5 = 1\)

### Step 2: Determine the exponent modulo 4
Since the pattern repeats every 4, we need to find \(51 \mod 4\):
\[ 51 \div 4 = 12 \quad \text{(remainder 3)} \]
Thus, \(51 \mod 4 = 3\).

### Step 3: Use the remainder to find the corresponding power of 2
From our earlier calculations, we know:
- \(2^1 \mod 5 = 2\)
- \(2^2 \mod 5 = 4\)
- \(2^3 \mod 5 = 3\)

Since \(51 \mod 4 = 3\), we have:
\[ 2^{51} \mod 5 = 2^3 \mod 5 = 3 \]

### Conclusion
The remainder when \(2^{51}\) is divided by 5 is **3**.

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