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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.

Which of the following is correct?

**A.** \(\ 2^{-1} \equiv 3 \pmod{7} \)

**B.** \(\ 3^{-1} \equiv 4 \pmod{7} \)

**C.** \(\ 5^{-1} \equiv 2 \pmod{7} \)

**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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