Number Theory

Modular Arithmetic Operations

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 92(mod7) 9 \equiv 2 \pmod{7} , what is 9×9×9×9×9(mod7)? 9 \times 9 \times 9 \times 9 \times 9 \pmod{7}?

What is 215(mod11)? 2 ^ {15} \pmod{11} ?

What is 534(mod31)? 5 ^ {34} \pmod{31} ?

Which of the following is congruent to 77777(mod10)? \large 7^{7^{7^{7^7}}} \pmod {10}?

What is 27(mod10)? 2 ^ {7} \pmod{10} ?

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