Units digit

24 $^{2017}$ $\times$ 36 $^{2017}$ $\times$ 17 $^{2017}$ = (24 $\times$ 36 $\times$ 17) $^{2017}$ = 14688 $^{2017}$

14688 (mod 10) = 8

2017 (mod 8) = 1

8 $^{1}$ = 8

Relevant wiki: Euler's Theorem

$\begin{aligned} 24^{2017}\times 36^{2017}\times 17^{2017} & \equiv (24\times 36\times 17)^{2017} \text{ (mod 10)} \\ & \equiv [(20+4)(30 + 6)(10+7)]^{2017} \text{ (mod 10)} \\ & \equiv [4\times 6 \times 7]^{2017} \text{ (mod 10)} \\ & \equiv [24 \times 7]^{2017} \text{ (mod 10)} \\ & \equiv [{\color{#3D99F6}4} \times 7]^{2017} \text{ (mod 10)} & \small \color{#3D99F6} \text{Note that }4^n \text{ mod }10 = \begin{cases} 4 & \text{if }n \text{ is odd.} \\ 6 & \text{if }n \text{ is even.} \end{cases} \\ & \equiv {\color{#3D99F6}4} \times 7^{\color{#D61F06} 2017 \text{ mod }\phi(10)} \text{ (mod 10)} & \small \color{#D61F06} \text{Since }\gcd (7, 10) = 1 \text{, Euler's theorem applies.} \\ & \equiv 4 \times 7^{\color{#D61F06} 2017 \text{ mod }4} \text{ (mod 10)} & \small \color{#D61F06} \text{Euler's totient function }\phi (10) = 4 \\ & \equiv 4 \times 7^{\color{#D61F06} 1} \text{ (mod 10)} \\ & \equiv 28 \equiv \boxed{8} \text{ (mod 10)} \end{aligned}$

$\begin{aligned} 24^{2017} \times 36^{2017} \times 17^{2017} \pmod{10} & \equiv \left(2{\color{#3D99F6}{4}} \times 3{\color{#3D99F6}{6}} \times 1{\color{#3D99F6}{7}}\right)^{2017} \pmod{10} \\ & \equiv {\color{#3D99F6}{8}}^{{\color{#D61F06}{2017 \ \text{mod} \ 4}}} \pmod{10} \\ & \equiv 8^{\color{#D61F06}{1}} \pmod{10} \\ & \equiv 8 \pmod{10} \end{aligned}$