find the remainder for $\color{purple}{TETRATION}$ such that $( {^{2021}2021}(\mod13))$

The only thing you need to know is that looking at a power $a^b$ modulo $k$ you can reduce the base $a$ modulo $k$ and the exponent $b$ modulo $\varphi(k)$.

Hence, when you consider the a tetration $$ {}^{2021}2021 = \color{purple}{2021}^{\color{blue}{2021}^{\color{orange}{2021}^{\cdots^{\color{red}{2021}}}}}, $$ you can reduce the base modulo $13$ the first exponent modulo $\varphi(13)=12$, the third exponent modulo $\varphi(12)=4$, where already $2021\equiv 1 \pmod 4$.

Hence $$ {}^{2021}2021 \equiv \color{purple}6^{\color{blue}5^{\color{orange}1^{\cdots^{\color{red}{2021}}}}} \equiv 6^5 \equiv 7776 \equiv 2 \pmod{13}. $$

Tags:

Modular Arithmetic

Elementary Number Theory

Solution Verification

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