When 100 is divided by a positive integer $x$, the remainder is 10. This means that $100-10=90$ is exactly divisible by $x$. It also means that $x$ is larger than 10, otherwise the remainder would be smaller than 10. Since 90 is exactly divisible by $x$, then $11 \times 90=990$ is also exactly divisible by $x$. Since $x>10$, then the next multiple of $x$ is $990+x$, which is larger than 1000. Thus, 990 is the largest multiple of $x$ less than 1000, and so the remainder when 1000 is divided by $x$ is $1000-990=10$.