jmc

Most of the numbers pair: $$\begin{array}{rl}& 1-2+3-4+5-6+7-8+9-10\\ & =(1-2)+(3-4)+(5-6)+(7-8)+(9-10)\\ & =-1-1-1-1-1+11\\ & =-5+11\\ & =6.\end{array}$$The sum is 6, so the remainder when this number is divided by 8 is $\boxed{6}$. (The quotient is 0.)