lean_workbook_plus_16446

We wish to calculate

We first note that $(1998,7) = (1999,7) = 1$ . By Fermat's Little Theorem, we have that $a^6 \equiv 1 \pmod{7}$ . \n\nWe have that $1998 = 6 \cdot 333$ from which we have that $1999^{1998} \equiv 1 \pmod{7}$ and since $1999 = 6 \cdot 333 +1$ that $1998^{1999} \equiv 1998 \pmod{7} \equiv 3 \pmod{7}$ \n\nThus we have $1998^{1999} + 1999^{1998} \equiv 1 +3 \pmod{7} \equiv 4 \pmod{7}$ .