Magic of 7

For $y= \left \lfloor 7 \left(\dfrac{\sqrt{|x| - 7} + \sqrt{7 - |x|}}{x + 7} + 7\right)^{700} \right \rfloor$ to be real, $|x| = 7$ else either $\sqrt{|x|-7}$ or $\sqrt{7-|x|}$ is unreal and for $\dfrac{\sqrt{|x| - 7} + \sqrt{7 - |x|}}{x + 7}$ to be defined, $x=7$ .

Therefore, we have:

$\begin{aligned} y & = \left \lfloor 7 \left(\dfrac{\sqrt{|x| - 7} + \sqrt{7 - |x|}}{x + 7} + 7\right)^{700} \right \rfloor \\ & = \left \lfloor 7 \left(\dfrac{\sqrt{0} + \sqrt{0}}{7 + 7} + 7\right)^{700} \right \rfloor \\ & = 7^{701} \\ \Rightarrow y & \equiv \color{#3D99F6}{7}^{701} \text{ (mod } \color{#3D99F6}{10}) \quad \quad \small \color{#3D99F6}{\text{As 7 and 10 are coprimes, we can apply Euler's totient theorem.}} \\ & \equiv 7^{\color{#3D99F6}{4} \times 175+1} \text{ (mod 10)} \quad \quad \small \color{#3D99F6}{7^{\phi (10)} \equiv 1 \text{ (mod 10) and totient function }\phi (10) = 4} \\ & \equiv \boxed{7} \text{ (mod 10)} \end{aligned}$