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Printjmc
number theory senior
Problem
Given that \begin{eqnarray}&(1)& x\text{ and }y\text{ are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=|x-y|. \end{eqnarray} How many distinct values of are possible?
Solution
We express the numbers as and . From this, we have\begin{eqnarray}z&=&|100a+10b+c-100c-10b-a|\\&=&|99a-99c|\\&=&99|a-c|\\ \end{eqnarray}Because and are digits, and and are both between 1 and 9 (from condition 1), there are possible values (since all digits except can be expressed this way).
Final answer
9