Find the remainder when `3^32` is divided by 50.

Options

A. 22
B. 41
C. 63
D. 88

Answer

To find the remainder when \( 3^{32} \) is divided by 50, we can use modular arithmetic and properties of exponents.

### Step-by-Step Solution:

1. **Identify the Problem**: We need to find \( 3^{32} \mod 50 \).

2. **Use Euler's Theorem**: Since 3 and 50 are coprime (they have no common factors other than 1), we can apply Euler's theorem. Euler's theorem states that if \( a \) and \( n \) are coprime, then:
\[ a^{\phi(n)} \equiv 1 \mod n \]
where \( \phi(n) \) is Euler's totient function.

3. **Calculate \( \phi(50) \)**:
- The prime factorization of 50 is \( 2 \times 5^2 \).
- Therefore,
\[ \phi(50) = 50 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{5}\right) = 50 \times \frac{1}{2} \times \frac{4}{5} = 20 \]

4. **Apply Euler's Theorem**: Since \( \phi(50) = 20 \), we have:
\[ 3^{20} \equiv 1 \mod 50 \]

5. **Reduce the Exponent**: Now, we can reduce the exponent 32 modulo 20:
\[ 32 \mod 20 = 12 \]
So, \( 3^{32} \equiv 3^{12} \mod 50 \).

6. **Calculate \( 3^{12} \mod 50 \)**:
- First, calculate \( 3^2 \):
\[ 3^2 = 9 \]
- Next, calculate \( 3^4 \):
\[ 3^4 = (3^2)^2 = 9^2 = 81 \equiv 31 \mod 50 \]
- Then, calculate \( 3^8 \):
\[ 3^8 = (3^4)^2 = 31^2 = 961 \equiv 11 \mod 50 \]
- Now combine \( 3^8 \) and \( 3^4 \) to find \( 3^{12} \):
\[ 3^{12} = 3^8 \times 3^4 \equiv 11 \times 31 \mod 50 \]
- Calculate \( 11 \times 31 \):
\[ 11 \times 31 = 341 \]
- Now reduce \( 341 \mod 50 \):
\[ 341 \div 50 = 6 \quad \text{(remainder 41)} \]
So, \( 341 \equiv 41 \mod 50 \).

7. **Final Result**: Therefore, the remainder when \( 3^{32} \) is divided by 50 is:
\[ \boxed{41} \]