If `3^(101)-2^(100)` is divided by 11, the remainder is

Answer

To find the remainder of \( 3^{101} - 2^{100} \) when divided by 11, we can use modular arithmetic.

### Step-by-Step Solution:

1. **Find \( 3^{101} \mod 11 \)**:
- We first notice that \( 3^5 \equiv 1 \mod 11 \) because \( 3^5 = 243 \) and \( 243 \div 11 = 22 \) remainder \( 1 \).
- Therefore, we can express \( 3^{101} \) in terms of \( 3^5 \):
\[ 3^{101} = 3^{100} \cdot 3 = (3^5)^{20} \cdot 3 \equiv 1^{20} \cdot 3 \equiv 3 \mod 11 \]

2. **Find \( 2^{100} \mod 11 \)**:
- Next, we check \( 2^5 \mod 11 \):
\[ 2^5 = 32 \quad \text{and} \quad 32 \div 11 = 2 \quad \text{remainder} \quad 10 \quad \text{(or } -1 \text{ mod 11)} \]
- Thus, \( 2^5 \equiv -1 \mod 11 \).
- We can express \( 2^{100} \) as:
\[ 2^{100} = (2^5)^{20} \equiv (-1)^{20} \equiv 1 \mod 11 \]

3. **Combine the results**:
- Now we can substitute back into our original expression:
\[ 3^{101} - 2^{100} \equiv 3 - 1 \mod 11 \]
- This simplifies to:
\[ 3 - 1 = 2 \mod 11 \]

4. **Final Result**:
- Thus, the remainder when \( 3^{101} - 2^{100} \) is divided by 11 is:
\[ \boxed{2} \]