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# Modular Arithmetic Operations

Considering the remainder "modulo" an integer is a powerful, foundational tool in Number Theory. You already use in clocks and work modulo 12.

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