Number Theory
Modular Arithmetic Operations
Given that \[ 2^{20} \equiv 1 \pmod{3} \quad \text{and} \quad 5^{10} \equiv 1 \pmod{3}, \] what is the remainder when \( 2^{20} \times 5^{10} \) is divided by 3?
What is \( 78 \times 49 \pmod{26} \)?
What is \(\underbrace{\left(6\cdot 6 \cdot 6 \cdots 6\right)}_{\text{15 6's}} \ \bmod{7}?\)
What is equivalent to
\[ 1!+2!+3!+ \cdots + 100! \pmod{12}? \]
Give your answer as an integer between 0 and 11, inclusive.
What is \( (10 \times 85) \pmod{5} ?\)