Fun with 2018 #10

$2018^4 \equiv 0 \text{ mod (4)}$ and $2018^4 \equiv 1 \text{ mod (25)}$ . This is a surprising property for powers of $2018$ . This implies that

a) $\forall \space n \in \mathbb{N}$ such that $n > 2$ , the last 2 digits of $2018^n$ are equal to the 2 last digits of $2018^{n + 4}$ , i. e, $2018^n$ only can end at $76, 68, 32$ or $24$ if $n > 2$ .

b) $\forall \space n \in \mathbb{N}$ such that $n > 3$ , the last 3 digits of $2018^n$ are equal to the 3 last digits of $2018^{n + 20}$ .

c) $\forall \space n \in \mathbb{N}$ such that $n > 4$ , the last 4 digits of $2018^n$ are equal to the 4 last digits of $2018^{n + 100}$ .

d) $\forall \space n \in \mathbb{N}$ such that $n > 5$ , the last 5 digits of $2018^n$ are equal to the 5 last digits of $2018^{n + 500}$ ...

On the other hand, the sequence $a_n$ has another surprising proprerty:

a) $\forall \space n \in \mathbb{N}$ such that $n > 3$ , the last 2 digits of $a_3$ which are $76$ are equal to the 2 last digits of $a_n$ , and the 2 last digits of $2018^{76}$ are $76$ , i.e, $2018^{76} \equiv 76 \text{ mod (100)}$

b) $\forall \space n \in \mathbb{N}$ such that $n > 4$ , the last 3 digits of $a_4$ which are $776$ are equal to the 3 last digits of $a_n$ , and the 3 last digits of $2018^{776}$ are $776$ , i.e, $2018^{776} \equiv 776 \text{ mod (1000)}$ .

c) $\forall \space n \in \mathbb{N}$ such that $n > 5$ , the last 4 digits of $a_5$ which are $9776$ are equal to the 4 last digits of $a_n$ , and the 4 last digits of $2018^{9776}$ are $9776$ , i.e, $2018^{9776} \equiv 9776 \text{ mod (10000)}$ .

d) $\forall \space n \in \mathbb{N}$ such that $n > 6$ , the last 5 digits of $a_6$ which are $79776$ are equal to the 5 last digits of $a_n$ , and the 5 last digits of $2018^{79776}$ are $79776$ , i.e, $2018^{79776} \equiv 79776 \text{ mod (100000)}$ .

e) $\forall \space n \in \mathbb{N}$ such that $n > 7$ , the last 6 digits of $a_7$ which are $379776$ are equal to the 6 last digits of $a_n$ , and the 6 last digits of $2018^{379776}$ are $379776$ , i.e, $2018^{379776} \equiv 379776 \text{ mod (1000000)}$ ....