The Power Difference

Relevant wiki: Euler's Theorem

Let $x = 10a + b$ , where $a$ and $b$ are positive integers. Then $x^{2017} - x \equiv (10a+b)^{2017} - (10a+b) \equiv (b^{2017}-b) \text{ (mod 10)}$ . There are two following cases.

Case 1: When $b$ and 10 are coprime integers, which are 1, 3, 7, and 9, then Euler's theorem applies.

$\begin{aligned} b^{2017} - b & \equiv (b^{\color{#3D99F6} 2017 \bmod \phi(10)} - b) \text{ (mod 10)} & \small \color{#3D99F6} \text{where }\phi (\cdot) \text{ is Euler's totient function} \\ & \equiv (b^{\color{#3D99F6} 2017 \bmod 4} - b) \text{ (mod 10)} \\ & \equiv b^1 - b \text{ (mod 10)} \\ & \equiv 0 \text{ (mod 10)} \end{aligned}$

Case 2: When $b$ and 10 are not coprime integers, which are 0, 2, 4, 5, 6 and 8.

$\begin{aligned} 0^{2017} - 0 & \equiv 0 \text{ (mod 10)} \\ 2^{2017} - 2 & \equiv 16^{504}\cdot 2 - 2 \equiv {\color{#3D99F6}6^{504}}\equiv {\color{#3D99F6}6}\cdot 2 - 2 \equiv 0 \text{ (mod 10)} & \small \color{#3D99F6} \text{Powers of 6 always end with 6.} \\ 4^{2017} - 4 & \equiv 16^{1008}\cdot 2 - 2 \equiv 6\cdot 2 - 2 \equiv 0 \text{ (mod 10)} \\ {\color{#3D99F6}5^{2017}} - 5 & \equiv {\color{#3D99F6}5}-5 \equiv 0 \text{ (mod 10)} & \small \color{#3D99F6} \text{Odd powers of 5 always end with 5.} \\ 6^{2017} - 6 & \equiv 0 \text{ (mod 10)} \\ 8^{2017} - 4 & \equiv 4096^{504}\cdot 2 - 2 \equiv 6\cdot 2 - 2 \equiv 0 \text{ (mod 10)} \end{aligned}$

Therefore, $x^{2017} - x \equiv \boxed 0 \text{ (mod 10)}$ for all integers $x \ge 0$ .

It is a fun fact worth knowing that the last digits of the powers $x,x^2,x^3,x^4,x^5,...$ repeat in a 4-cycle for all positive integers $x$ . (It takes just one minute to verify this directly, or one can invoke some theory.) Going through this cycle 504 times, we see that the last digits of $x$ and $x^{1+504\times 4}=x^{2017}$ are the same. Thus $x^{2017}-x \equiv \boxed{0} \text{ (mod 10)}$ . More generally, $x^{4n+1}\equiv x \text{ (mod 10)}$ for all positive integers $n$ .

https://brilliant.org/wiki/finding-the-last-digit-of-a-power/