2002 AIME II Problems/Problem 1

Problem

Given that

$\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\

   &(2)& \text{y is the number formed by reversing the digits of x; and}\\

&(3)& z=|x-y|. \end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

How many distinct values of $z$ are possible?

We express the numbers as $x=100a+10b+c$ and $y=100c+10b+a$ . From this, we have

$\begin{eqnarray*}z&=&|100a+10b+c-100c-10b-a|\\&=&|99a-99c|\\&=&99|a-c|\\ \end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

Because $a$ and $c$ are digits, and $a$ is between 1 and 9, there are $\boxed{9}$ possible values.

2002 AIME II (Problems • Answer Key • Resources)
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