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Considering the remainder "modulo" an integer is a powerful, foundational tool in Number Theory. You already use in clocks and work modulo 12.

Given that \( 9 \equiv 2 \pmod{7} \), what is

\[ 9 \times 9 \times 9 \times 9 \times 9 \pmod{7}? \]

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

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

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

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

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