The units digit of a power tower of consecutive numbers, from 2019 to 1

HINT

$2019\equiv -1\pmod{10}\Rightarrow 2019^a\equiv (-1)^a\pmod{10}$


The unit digit of a number $N$ can be computed as the class of $N$ modulo $10$ and taking its representant in $\{0,1,...,9\}$

You're asking for the units digit of a number of the form $2019^M$. Since the class modulo $10$ of $2019$ is $-\overline1$, you are in fact computing $(-1)^M\bmod 10$.

The result is $1$ if $M$ is even and $9$ if $M$ is odd.

Since your actual $M$ is of the form $M=2018^m$ and $2018$ is even, the former case holds.


$2019^{2018^{2017^{.^{.^{.^{3^{2^{1}}}}}}}}\equiv9^{2018^{2017^{.^{.^{.^{3^{2^{1}}}}}}}}\pmod{10}$

As $\phi(10)=4,(9,10)=1$ and the exponent is multiple of $4$

$9^{2018^{2017^{.^{.^{.^{3^{2^{1}}}}}}}}\equiv9^0\pmod{10}$

Tags:

Exponentiation

Elementary Number Theory

Power Towers

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