9. $[7]$ Find the remainder when $1^{2}+3^{2}+5^{2}+\cdots+99^{2}$ is divided by 1000 .
Answer: 650 We have $S=\sum_{i=0}^{49}(2 i+1)^{2}=\sum_{i=0}^{49} 4 i^{2}+4 i+1=4 \cdot \frac{49 \cdot 50 \cdot 99}{6}+4 \cdot \frac{49 \cdot 50}{2}+50 \equiv$ $700+900+50(\bmod 1000) \equiv 650(\bmod 1000)$