Number Theory

# Modular Arithmetic - Multiplication

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

$1!+2!+3!+ \cdots + 100! \pmod{12}?$

What is $(10 \times 85) \pmod{5} ?$