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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.

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

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

Give your answer as an integer between 0 and 11, inclusive.

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

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