The remainder when \(7^{103} + 7^{101}\) is divided by \(9\) is __________

The remainder is $2$. Since $7^3\equiv1\pmod9$, the exponents $103$ and $101$ reduce to $1$ and $2$ in the cycle, giving residues $7$ and $4$. Their sum $11$ leaves remainder $2$, whereas treating $101$ as a multiple of three incorrectly gives $7+1=8$.