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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 - Exponentiation

As we've now covered modular addition and modular multiplication (repeated modular addition), the next operation to consider is repeated modular multiplication, aka modular exponentiation!

Given that $$9 \equiv 2 \pmod{7}$$, what is $9 \times 9 \times 9 \times 9 \times 9 \pmod{7}?$

What is $$2 ^ {15} \pmod{11} ?$$

What is $$5 ^ {34} \pmod{31} ?$$

Which of the following is congruent to $\large 7^{7^{7^{7^7}}} \pmod {10}?$

What is $$2 ^ {7} \pmod{10} ?$$

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