The remainder when `3^21` is divided by 5 is

Options

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

Answer

To find the remainder when \(3^{21}\) is divided by 5, we can use the concept of modular arithmetic, specifically focusing on the pattern of powers of 3 modulo 5.

### Step-by-Step Solution:

1. **Calculate the first few powers of 3 modulo 5:**
- \(3^1 \mod 5 = 3\)
- \(3^2 \mod 5 = 9 \mod 5 = 4\)
- \(3^3 \mod 5 = 27 \mod 5 = 2\)
- \(3^4 \mod 5 = 81 \mod 5 = 1\)

2. **Observe the pattern:**
- The results of \(3^n \mod 5\) are as follows:
- \(3^1 \equiv 3\)
- \(3^2 \equiv 4\)
- \(3^3 \equiv 2\)
- \(3^4 \equiv 1\)
- Notice that after \(3^4\), the pattern repeats every 4 terms: \(3, 4, 2, 1\).

3. **Determine the position of \(3^{21}\) in the cycle:**
- Since the pattern repeats every 4 terms, we can find the equivalent exponent by calculating \(21 \mod 4\):
\[ 21 \div 4 = 5 \quad \text{(quotient)} \]
\[ 21 - (4 \times 5) = 1 \quad \text{(remainder)} \]
- Thus, \(21 \mod 4 = 1\).

4. **Find the corresponding value from the pattern:**
- From our earlier calculations, we see that \(3^1 \equiv 3 \mod 5\).

5. **Conclusion:**
- Therefore, the remainder when \(3^{21}\) is divided by 5 is \(3\).

### Final Answer:
The remainder when \(3^{21}\) is divided by 5 is **3**.