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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.

# Multiplicative Inverses

Which of the following is correct?

$$\text{(A) } 2^{-1} \equiv 3 \pmod{7}$$
$$\text{(B) } 3^{-1} \equiv 4 \pmod{7}$$
$$\text{(C) } 5^{-1} \equiv 2 \pmod{7}$$
$$\text{(D) } 6^{-1} \equiv 6 \pmod{7}$$

What is $$5 ^ {-1} \pmod{17} ?$$

Hint: Remember that inverses multiply to 1.

What is

$\large 14^{-1} \pmod{17} ?$

What is $\large 10! \pmod{11}?$

What is $$2 ^ {-1} \pmod{39} ?$$

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