When 100 is divided by 12 , the remainder is 4 .

When 100 is divided by a positive integer $x$, the remainder is 10 .

When 1000 is divided by $x$, the remainder is
(A) 10
(B) 100
(C) 0
(D) 1
(E) 90
Solution 1

When a number $n$ is divided by $x$, the remainder is the difference between $n$ and the largest multiple of $x$ less than $n$. 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 $1000-990=10$.

## Solution 2

When 100 is divided by a positive integer $x$, the remainder is 10 . The remainder is the difference between 100 and the largest multiple of $x$ less than 100 .

Therefore, the largest multiple of $x$ less than 100 is $100-10=90$.

It also means that $x$ is larger than 10 , otherwise the remainder would be smaller than 10 .

We can choose $x=15$, since $6 \times 15=90$ and 15 is larger than 10 .

What is the remainder when 1000 is divided by 15 ? Using a calculator, $1000 \div 15=66.666 \ldots$ and $66 \times 15=990$.

Thus, the difference between 1000 and the largest multiple of 15 less than 1000 (that is, 990) equals 10 and so the remainder is equal to 10 .

ANSWER: (A)