Divisibility test 3

Since 15, 23 and 19 are coprime integers, we can apply Euler's theorem as follows:

$\begin{aligned} 15^{23} + 23^{123} & \equiv 15^{\color{#3D99F6}23 \text{ mod }\phi(19)}+23^{\color{#3D99F6}123 \text{ mod }\phi(19)} & \small \color{#3D99F6} \text{Euler's totient function }\phi(19) = 18 \\ & \equiv 15^5+23^{15} \text{ (mod 19)} \\ & \equiv (19-4)^5+(19+4)^{15} \text{ (mod 19)} \\ & \equiv (-4)^5+4^{15} \text{ (mod 19)} \\ & \equiv (16)(16)(-4)+2^{30 \text{ mod }18} \text{ (mod 19)} \\ & \equiv (-3)(-3)(-4)+2^{12} \text{ (mod 19)} \\ & \equiv -36+16^3 \text{ (mod 19)} \\ & \equiv 2+(-3)^3 \text{ (mod 19)} \\ & \equiv 2-27 \text{ (mod 19)} \\ & \equiv -25 \text{ (mod 19)} \\ & \equiv \boxed{13} \text{ (mod 19)} \end{aligned}$