Remainder of Power Expression Modulo 7

$19 \equiv 5 \equiv -2 \pmod 7$. $19^{20} \equiv (-2)^{20} = 2^{20} \pmod 7$. Since $2^3 \equiv 1 \pmod 7$, $2^{20} = (2^3)^6 \cdot 2^2 \equiv 1^6 \cdot 4 = 4 \pmod 7$.

$20 \equiv 6 \equiv -1 \pmod 7$. $20^{19} \equiv (-1)^{19} = -1 \equiv 6 \pmod 7$.

$19^{20} - 20^{19} \equiv 4 - (-1) = 5 \pmod 7$. Remainder is 5.